#7604138

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1296 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:

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

Polycyclic presentation:

2^9|a^2=fgh,c^2=h,d^2=g,e^2=fi,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, 7511334);