#6461803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1505 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)(3,6)(4,8)(5,7)(17,18,22,27,23,28,21,26)(19,29,30,32,20,25,31,24);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,23)(21,30)(22,31)(25,27)(26,29)(28,32);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19,23,20)(18,25,28,29)(21,31,22,30)(24,26,32,27);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(17,20,23,19)(18,25,28,29)(21,30,22,31)(24,26,32,27);(17,21,23,22)(18,26,28,27)(19,30,20,31)(24,29,32,25)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368999);