#72300941

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 8683 conjugacy classes of subgroups; only the 2989 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=h,d^2=hij,b^a=bi,c^b=ch,d^b=dg,d^c=dhj,e^c=eh,e^d=eh,f^a=fhij,f^b=fgh,f^c=fij,f^e=fgh

Conjugacy classes & character table

112 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, 3, F.8);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);