#6624266

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1305 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^c=ei,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531462);