#6559792

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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

1274 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6466988);