Finite Group Databasebeta

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#72287315

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#130204
commutatorC2×M32C22×C4
Frattini, cosocleC2×#504C23
prime radical#7624863C2
LCSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #33434 ◁ G
LECSG ▷ C2×#504 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

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

normal subgroups only

This group has 835 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(2,9,15,34)(3,16)(4,21)(5,26)(6,27)(7,29)(10,12,31,44)(11,38,43,63)(13,33,28,56)(14,32,46,37)(17,49)(18,45)(19,52)(20,48)(22,54)(23,24)(25,55)(30,57,60,42)(36,53,58,41)(39,40,59,64)(47,51)(61,62);(2,10)(3,16)(4,21)(5,26)(6,27)(7,29)(9,12)(11,39)(13,37)(14,33)(15,31)(17,49)(18,45)(19,52)(20,48)(22,54)(23,24)(25,55)(28,32)(30,53)(34,44)(36,42)(38,40)(41,60)(43,59)(46,56)(47,51)(57,58)(61,62)(63,64);(1,2,3,11,6,28,19,53,26,44,52,64,7,14,16,42,8,15,20,43,29,13,45,41,5,12,18,40,27,46,48,57)(4,9,17,30,21,37,51,39,22,31,50,58,23,33,47,38,25,34,49,60,55,32,61,59,24,10,35,36,54,56,62,63)

Polycyclic presentation:

2^10|a^2=ghj,c^2=d,d^2=hi,e^2=h,f^2=j,g^2=ij,h^2=i,i^2=j,c^a=ce,c^b=cefj,d^a=dh,d^b=dh,e^a=ehi,e^b=eg,f^a=fgh,f^b=fgh,f^e=fj,g^a=gi,g^b=gi,g^c=gj,h^a=hij,h^b=hij,i^a=ij,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72287373, #72320889, #72342911
maximal normal subgroup of#72287373, #72320889, #72342911

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.7*F.8*F.10);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);