#72304097

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 8486 conjugacy classes of subgroups; only the 3097 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=j,d^2=h,c^a=ci,d^b=dg,d^c=dg,e^a=eh,e^b=ej,e^c=eg,e^d=eh,f^a=fi,f^d=fgh,f^e=fij

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