#6559812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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

1242 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)(19,24)(20,28)(21,25)(22,32)(23,26)(27,31)(29,30);(1,3,4,8)(2,5,6,7)(9,11,15,13)(10,16,14,12)(17,19,23,29)(18,24,26,30)(20,25,31,32)(21,27,22,28);(9,12,15,16)(10,11,14,13)(17,20,22,24,23,31,21,30)(18,19,25,28,26,29,32,27);(9,13,15,11)(10,16,14,12)(17,21,23,22)(18,25,26,32)(19,27,29,28)(20,24,31,30);(9,14,15,10)(11,12,13,16)(17,21,23,22)(18,25,26,32)(19,28,29,27)(20,30,31,24)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,d^c=di,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, 6467008);