#483759

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=g,e^2=g,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=dgi,e^a=eg,e^b=eg,d^c=dgi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390955);