#6624298

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6531494);