#7006227

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

2061 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)(16,27,30,35,29,38,24,20)(17,21,25,36,32,37,31,33)(28,40,41,44,42,43,39,34);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,22,23)(16,24,29,30)(17,31,32,25)(20,27,35,38)(21,36,37,33)(28,41,42,39)(34,43,44,40);(5,7)(6,9)(8,12)(10,11)(13,16,18,29)(14,20,22,35)(15,24,26,30)(17,28,32,42)(19,27,23,38)(21,34,37,44)(25,39,31,41)(33,40,36,43);(1,3,4,2)(7,12)(9,11)(13,17,18,32)(14,21,22,37)(15,25,26,31)(16,28,29,42)(19,33,23,36)(20,34,35,44)(24,39,30,41)(27,40,38,43);(6,10)(9,11)(17,32)(21,37)(25,31)(28,42)(33,36)(34,44)(39,41)(40,43)

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=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913423);