#72330799

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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 6260 conjugacy classes of subgroups; only the 3669 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:

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

Polycyclic presentation:

2^10|a^2=i,b^a=bij,c^a=ci,c^b=cg,d^a=dgi,d^b=dgi,f^b=fgi,f^c=fi,f^d=fh,f^e=fgi

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72330942
maximal normal subgroup of	#72330942
maximal quotient of	#72333204

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