Finite Group Databasebeta

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#72321632

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#152418
commutatorC2×C32C22×C4
Frattini, cosocleC22×C32C23
prime radicalC2×#41D16
LCSG ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ C2×#2258 ◁ G
LECSG ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

323 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 128
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 128 points:

(1,2,7,14)(3,18)(4,24)(5,29,6,37)(8,46,43,62)(9,51,44,84)(10,54,45,17)(11,20)(12,25)(13,40,63,34)(15,32,66,33)(16,71,67,49)(19,56)(21,81)(22,83)(23,77)(26,94)(27,79)(28,58)(30,74)(31,69)(35,108,41,118)(36,86,42,90)(38,68)(39,87)(47,60)(48,88)(50,125,106,95)(52,75)(53,89)(55,76)(57,113)(59,102)(61,92)(64,127,126,128)(65,91,114,85)(70,96,122,105)(72,80)(73,93)(78,110)(82,119)(97,109)(98,103,111,104)(99,117,112,107)(100,121)(101,124)(115,120)(116,123);(1,3,7,23)(2,11)(4,17)(5,30,6,38)(8,47,43,78)(9,52,44,120)(10,55,45,12)(13,60)(14,56)(15,68)(16,72)(18,29)(19,40,20,34)(21,32,82,33)(22,71,73,49)(24,85,58,91)(25,90)(26,95)(27,67)(28,54)(31,65)(35,109,41,94)(36,88,42,61)(37,77)(39,114)(46,57)(48,70)(50,87,106,69)(51,53)(59,96,123,105)(62,124)(63,110)(64,116)(66,74)(75,108)(76,86)(79,127,80,128)(81,98)(83,99)(84,121)(89,104,100,103)(92,122)(93,112)(97,125)(101,117,113,107)(102,126)(111,119)(115,118);(1,4,10,56,7,28,45,11)(2,12,17,3,14,55,54,23)(5,31,36,110,6,39,42,60)(8,48,50,74,43,92,106,68)(9,53,49,27,44,121,71,72)(13,61,65,30,63,88,114,38)(15,69,70,47,66,87,122,78)(16,73,51,52,67,22,84,120)(18,34,25,91,77,40,76,85)(19,37,24,86,20,29,58,90)(21,62,123,125,82,46,59,95)(26,96,124,32,97,105,57,33)(35,81,107,102,41,119,117,116)(64,113,98,109,126,101,111,94)(75,128,93,103,115,127,83,104)(79,118,89,99,80,108,100,112)

Polycyclic presentation:

2^10|a^2=f,b^2=degj,c^2=fij,d^2=ghj,e^2=ghj,g^2=hi,h^2=ij,i^2=j,b^a=bd,c^a=cdj,c^b=cdefgij,d^a=dg,d^b=dg,d^c=dg,e^a=egj,e^b=eg,e^c=egj,f^b=fj,f^c=fj,g^a=gh,g^b=gh,g^c=gh,h^a=hi,h^b=hi,h^c=hi,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72321641, #72322933, #72322982
maximal normal subgroup of#72321641, #72322933, #72322982

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.4*F.5*F.7*F.10);;
gap> SetPower(coll, 3, F.6*F.9*F.10);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);