#507255

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

546 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(18,23)(19,26)(25,30)(28,32);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,18,24,30,29,19,27,32,20,23,21,25,22,26,31,28);(10,13)(14,15)(17,19,20,26)(18,22,23,29)(21,25,24,30)(27,28,31,32);(2,6)(5,7)(9,11,12,16)(10,14,13,15)

Polycyclic presentation:

2^9|b^2=efgi,c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gh,h^c=hi

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, 414451);