#72330591

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 7664 conjugacy classes of subgroups; only the 1055 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=h,c^2=fg,d^2=g,b^a=bfi,c^a=cg,c^b=cfgj,d^a=dfg,d^b=dij,d^c=df,e^a=eij,e^b=efh,e^c=efgj,e^d=efgij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

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