Finite Group Databasebeta

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#72284201

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139486
commutatorC2×#117C22×C4
Frattini, cosocle#2309C23
prime radicalC22×H(5,𝔽2)D4
LCSG ▷ C2×#117 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35616 ◁ G
LECSG ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1368 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 1368 conjugacy classes of subgroups — too many to draw. Only its 77 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fj,c^2=g,d^2=ghi,e^2=ghi,b^a=bj,c^a=cd,c^b=ce,d^a=dg,d^b=dh,d^c=dgh,e^a=ehj,e^b=eg,e^c=eghj,e^d=ej,f^c=fhi,f^d=fi,f^e=fij,g^a=gi,g^b=gij,g^d=gj,g^e=gj,h^a=hij,h^b=hi,h^c=hj,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284293, #72285872
maximal normal subgroup of#72284293, #72285872

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