#7006355

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2027 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,15,23,18,22,26,19)(16,27,30,34,29,37,24,20)(17,21,25,36,31,35,39,32)(28,40,42,43,41,44,38,33);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,22,23)(16,24,29,30)(17,25,31,39)(20,27,34,37)(21,32,35,36)(28,38,41,42)(33,40,43,44);(5,7)(6,9)(8,12)(10,11)(13,16,18,29)(14,20,22,34)(15,24,26,30)(17,28,31,41)(19,27,23,37)(21,33,35,43)(25,38,39,42)(32,40,36,44);(1,3,4,2)(7,12)(9,11)(13,17,18,31)(14,21,22,35)(15,25,26,39)(16,28,29,41)(19,32,23,36)(20,33,34,43)(24,38,30,42)(27,40,37,44);(6,10)(9,11)(17,31)(21,35)(25,39)(28,41)(32,36)(33,43)(38,42)(40,44)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913551);