#506137

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ehi,c^2=e,d^2=f,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413333);