#6461638

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1704 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)(4,5)(7,8)(17,18,23,20)(19,26,27,32)(21,25,22,31)(24,28,30,29);(9,10)(11,15)(12,14)(13,16)(17,19,22,29,23,27,21,28)(18,24,25,26,20,30,31,32);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,23,18)(19,24,27,30)(21,25,22,31)(26,28,32,29);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(19,27)(24,30)(26,32)(28,29);(17,21,23,22)(18,25,20,31)(19,28,27,29)(24,26,30,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=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, 6368834);