#6558962

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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)(17,18)(19,29)(20,32)(21,26)(22,27)(23,28)(24,31)(25,30);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,31,23,32,22,25)(18,24,27,29,28,30,26,20);(10,13)(14,16)(17,20,22,30,23,29,21,24)(18,25,26,32,28,31,27,19);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,18)(19,30)(20,31)(21,27)(22,26)(23,28)(24,32)(25,29);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,25,32,31)(20,24,29,30)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=fg,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6466158);