#7590421

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	#45066	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,22,24)(19,26,29,31)(20,28,23,32)(21,27,30,25);(3,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,28,22,29,30,32)(18,23,25,26,24,20,27,31);(1,3,4,8)(2,5,6,7)(17,18)(19,20)(21,25)(22,24)(23,29)(26,32)(27,30)(28,31);(2,6)(5,7)(11,15)(13,16)(18,24)(20,23)(25,27)(26,31);(17,20,30,26,22,23,21,31)(18,19,27,32,24,29,25,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7497617);