#502176

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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

776 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)(9,10,14,20)(11,21,15,17)(12,24,25,35)(13,16,29,31)(18,34,36,26)(19,23,32,27)(22,39,30,37)(28,38,40,33);(1,2)(3,4)(5,7,8,6)(10,16)(11,15)(12,25)(13,27)(17,32)(18,35)(19,21)(20,31)(23,29)(24,36)(26,40)(28,34)(33,37)(38,39);(2,4)(6,7)(9,11,14,15)(10,17,20,21)(12,22,25,30)(13,23,29,27)(16,19,31,32)(18,33,36,38)(24,37,35,39)(26,28,34,40);(9,12,14,25)(10,18,20,36)(11,22,15,30)(13,28,29,40)(16,24,31,35)(17,33,21,38)(19,37,32,39)(23,34,27,26)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,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,d^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409372);