#7601207

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1325 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=hi,d^2=gi,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7508403);