#72232687

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 9348 conjugacy classes of subgroups; only the 3073 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=h,c^2=g,d^2=h,e^2=j,f^2=hi,c^a=cg,c^b=ci,e^a=ehj,e^d=ehj,f^a=fgi,f^b=fgh,f^c=fij,f^d=fj,f^e=fgh

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232710
maximal normal subgroup of	#72232710

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