#510068

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

663 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)(11,13)(12,14)(15,16)(17,18,20,22)(19,25,27,23)(21,24,28,29)(26,31,32,30);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(18,22)(19,26)(21,29)(23,31)(24,28)(25,30)(27,32);(10,14)(13,16)(17,19)(18,23)(20,27)(21,30)(22,25)(24,32)(26,29)(28,31);(2,6)(3,8)(10,14)(13,16)(17,18,20,22)(19,23,27,25)(21,24,28,29)(26,30,32,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,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, 417264);