#502162

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	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

790 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)(9,10,14,19)(11,20,23,38)(12,15,27,31)(13,28,26,18)(16,32,34,21)(17,35,30,24)(22,39,29,36)(25,37,40,33);(1,2)(3,4)(5,7,8,6)(10,15)(12,24)(13,26)(16,20)(17,18)(19,31)(21,25)(22,29)(27,35)(28,30)(32,40)(33,39)(34,38)(36,37);(2,4)(6,7);(9,11,14,23)(10,16,19,34)(12,25,27,40)(13,29,26,22)(15,20,31,38)(17,36,30,39)(18,37,28,33)(21,35,32,24)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,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, 409358);