#7594221

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,13,15,16)(17,18,22,24)(19,25,26,32)(20,21,28,29)(23,30,31,27);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,22,24)(19,25,26,32)(20,21,28,29)(23,30,31,27);(1,3,5,4)(2,7,6,8)(17,19,21,27,22,26,29,30)(18,23,20,25,24,31,28,32);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,21,24,22,28,29,18)(19,25,27,31,26,32,30,23);(1,4,5,3)(2,7,6,8)(19,26)(23,31)(25,32)(27,30)

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,c^2=fg,d^2=fg,e^2=f,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, 7501417);