#502166

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(5,6,8,7)(9,10)(11,20)(12,15)(13,16)(14,28)(17,21)(18,24)(19,26)(22,36)(23,39)(25,35)(27,30)(29,34)(31,37)(32,38)(33,40);(1,2)(3,4)(5,7,8,6)(10,15)(12,24)(14,26)(16,30)(17,20)(18,19)(21,25)(23,29)(27,37)(28,31)(32,36)(33,39)(34,35)(38,40);(2,4)(6,7)(10,16)(15,30)(17,32)(18,31)(19,28)(20,36)(33,35)(34,39);(9,11,13,22)(10,17,16,32)(12,25,27,40)(14,29,26,23)(15,20,30,36)(18,34,31,39)(19,35,28,33)(21,37,38,24)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409362);