#502156

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	C₂²×#135	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

779 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)(11,17)(12,23)(13,16)(14,20)(15,29)(18,24)(19,26)(21,37)(22,31)(25,39)(27,33)(28,32)(30,36)(34,38)(35,40);(1,2)(3,4)(5,7,8,6)(10,16)(11,15)(13,26)(17,19)(18,23)(20,32)(21,30)(22,28)(24,27)(29,31)(33,36)(34,39)(35,37)(38,40);(2,4)(6,7)(9,11,14,15)(10,17,20,29)(12,21,25,30)(13,22,28,26)(16,31,32,19)(18,33,34,35)(23,37,39,36)(24,27,38,40);(9,12,14,25)(10,18,20,34)(11,21,15,30)(13,27,28,40)(16,23,32,39)(17,33,29,35)(19,36,31,37)(22,38,26,24)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,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, 409352);