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#72227776

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC23C2×B(4,𝔽2)
commutator, Frattini, cosocleC22×C22⋊C4C24
prime radicalC27D4
LCSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

6609 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 6609 conjugacy classes of subgroups — too many to draw. Only its 217 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) = 32
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32);(1,2,4,6)(3,5,7,8)(10,14)(11,12)(15,16)(18,22)(19,25)(23,30)(24,27)(26,32)(29,31);(1,3)(2,5)(4,7)(6,8)(9,10,13,15)(11,14,12,16)(17,19)(18,23)(20,25)(21,26)(22,29)(24,31)(27,30)(28,32);(3,7)(5,8)(9,11,13,12)(10,14,15,16)(17,20)(18,24)(19,25)(21,28)(22,27)(23,31)(26,32)(29,30)

Polycyclic presentation:

2^10|b^2=g,c^2=i,d^2=i,e^2=i,b^a=bj,c^a=cf,c^b=ce,d^a=dh,d^b=di,d^c=dg,e^a=eh,e^b=ei,e^c=ei,f^b=fhj,f^d=fj,f^e=fj,h^c=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227783
maximal normal subgroup of#72227783

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