#72304108

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 7744 conjugacy classes of subgroups; only the 2985 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=h,d^2=j,e^2=i,c^b=ch,d^c=dgh,e^a=egi,e^b=eghij,e^d=ej,f^a=fij,f^c=fgj,f^e=fij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304145
maximal normal subgroup of	#72304145

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