#72287763

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₈	C₂⁴
commutator, socle	C₂⁴	C₂³×C₈
prime radical	C₂³×#41	C₈
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=efij,d^2=h,f^2=i,i^2=j,b^a=bh,c^a=ce,c^b=cg,d^c=dj

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287806, #72287826
maximal normal subgroup of	#72287806, #72287826

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