#502003

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,9)(8,10)(11,12)(13,14)(15,25)(16,30)(17,33)(18,21)(19,36)(20,28)(22,31)(23,27)(24,34)(26,39)(29,44)(32,41)(35,37)(38,42)(40,43);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,26,42,19,29,17,28,18,16,27,43,34,32,35,31)(14,20,37,44,24,38,23,25,21,22,33,41,36,40,39,30);(2,4)(6,10)(9,12)(14,21)(20,22)(23,39)(24,36)(25,30)(33,37)(38,40)(41,44);(13,16,26,43,19,32,17,31,18,15,27,42,34,29,35,28)(14,22,37,41,24,40,23,30,21,20,33,44,36,38,39,25)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409199);