#7613709

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1995 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520905);