#6678366

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1556 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,23,28,26)(21,24,30,29)(25,31,27,32);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19,22,28)(18,23,20,26)(21,25,30,27)(24,31,29,32);(2,6)(5,8)(9,11,12,15)(10,14,13,16)(19,25)(21,30)(23,31)(24,29)(26,32)(27,28);(10,13)(14,16)(17,20,22,18)(19,26,28,23)(21,29,30,24)(25,32,27,31);(19,25)(21,30)(23,31)(24,29)(26,32)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

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, 6585562);