#359368

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266564);