#6570289

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39262	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 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)(17,18,22,25)(19,23,27,31)(20,29,24,21)(26,28,30,32);(1,3,5,4)(2,6,8,7)(9,11,15,13)(10,16,14,12)(17,19,22,27)(18,23,25,31)(20,30,24,26)(21,28,29,32);(9,12,15,16)(10,11,14,13)(17,20,21,18,22,24,29,25)(19,26,28,31,27,30,32,23);(1,3,5,4)(2,7,8,6)(9,13,15,11)(10,16,14,12)(19,27)(23,31)(26,30)(28,32);(1,4,5,3)(2,6,8,7)(9,14,15,10)(11,12,13,16)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,d^2=f,e^2=f,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, 6477485);