#72263202

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 8129 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:

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

Polycyclic presentation:

2^10|b^2=h,d^2=hi,e^2=ij,d^b=dg,d^c=dgj,e^a=ehij,e^b=eg,e^c=ej,e^d=egi,f^a=fhi,f^b=fh,f^d=fgj,f^e=fij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263231
maximal normal subgroup of	#72263231

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