#6570170

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1545 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)(3,7)(4,6)(5,8)(9,10,15,14)(11,12,13,16)(17,18,23,26)(19,24,29,31)(20,21,25,22)(27,30,32,28);(1,3,5,4)(2,6,8,7)(9,11,15,13)(10,16,14,12)(17,19)(18,24)(20,27)(21,30)(22,28)(23,29)(25,32)(26,31);(9,12,15,16)(10,11,14,13)(17,20,22,18,23,25,21,26)(19,27,28,24,29,32,30,31);(1,3,5,4)(2,7,8,6)(9,13,15,11)(10,16,14,12)(17,21,23,22)(18,25,26,20)(19,28,29,30)(24,27,31,32);(1,4,5,3)(2,6,8,7)(9,14,15,10)(11,12,13,16)(19,29)(24,31)(27,32)(28,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=g,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6477366);