#7601215

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

1318 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)(17,18,23,19)(20,27,28,25)(21,30,31,32)(22,29,24,26);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,14)(13,15)(17,19,22,29,23,18,24,26)(20,25,30,31,28,27,32,21);(1,3,4,7)(2,5,6,8)(9,11,13,12)(10,14,15,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,32)(26,31);(2,6)(5,8)(9,11,13,12)(10,14,15,16)(18,19)(20,28)(26,29)(30,32);(9,12,13,11)(10,14,15,16)(17,21,24,27,23,31,22,25)(18,20,29,32,19,28,26,30)

Polycyclic presentation:

2^9|a^2=i,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, 7508411);