#7064235

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

1860 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)(13,14,19,21,18,23,17,24)(15,20,30,35,25,34,27,32)(16,22,31,36,28,38,29,37)(26,33,41,43,39,44,40,42);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,25)(14,20,23,34)(16,26,28,39)(17,27,19,30)(21,35,24,32)(22,33,38,44)(29,40,31,41)(36,43,37,42);(5,7,8,12)(6,9,10,11)(14,21)(15,25)(16,28)(17,19)(20,32)(22,37)(23,24)(33,43)(34,35)(36,38)(40,41)(42,44);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,26)(17,29)(18,28)(19,31)(20,33)(21,36)(23,38)(24,37)(25,39)(27,40)(30,41)(32,42)(34,44)(35,43);(6,10)(9,11)(14,23)(15,25)(16,28)(21,24)(27,30)(29,31)(33,44)(42,43)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=hi,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, 6971431);