#6520559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2278 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,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, 6427755);