#72330397

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

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 7759 conjugacy classes of subgroups; only the 1111 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)(11,12)(13,14)(15,16)(17,18)(19,23)(20,21)(22,24);(1,2)(3,4)(5,6,7,8)(10,14)(12,16)(17,18)(19,23)(20,21)(22,24)(27,31)(29,32);(10,14)(11,15)(18,21)(19,24)(25,26,28,30)(27,29,31,32);(2,4)(6,8)(9,11)(10,12)(13,15)(14,16)(17,18)(19,23)(20,21)(22,24)(25,26)(27,32)(28,30)(29,31);(1,2)(3,4)(5,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,29)(28,31)(30,32)

Polycyclic presentation:

2^10|b^2=h,c^2=fj,b^a=bi,c^a=cg,c^b=cf,d^b=dfgi,d^c=dfgj,e^a=efi,e^b=efghj,e^c=ef,e^d=eij

Conjugacy classes & character table

142 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, 2, F.8);;
gap> SetPower(coll, 3, F.6*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);