#7613699

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
prime radical	#45385	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1975 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)(13,14,20,24)(15,21,29,26)(16,30,33,32)(17,19,23,18)(22,35,40,34)(25,39,42,44)(27,38,28,36)(31,37,43,41);(2,4)(6,9)(10,12)(13,15,20,29)(14,21,24,26)(16,31,33,43)(17,36,23,38)(18,28,19,27)(22,39,40,44)(25,35,42,34)(30,37,32,41);(5,7,8,11)(6,10,9,12)(13,16,19,35,20,33,18,34)(14,22,17,32,24,40,23,30)(15,25,28,43,29,42,27,31)(21,37,38,44,26,41,36,39);(1,2,3,4)(13,17,18,14,20,23,19,24)(15,26,27,38,29,21,28,36)(16,32,34,22,33,30,35,40)(25,41,31,44,42,37,43,39);(6,9)(10,12)(16,33)(22,40)(25,42)(30,32)(31,43)(34,35)(37,41)(39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=hi,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,e^c=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, 7520895);