#509305

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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

612 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)(7,8)(9,11)(10,12)(13,14)(15,20)(16,27)(17,21)(18,24)(19,33)(22,28)(23,30)(25,41)(26,36)(29,43)(31,35)(32,37)(34,40)(38,42)(39,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,19)(14,20,24,33)(16,25,29,34)(17,26,32,30)(21,36,37,23)(22,35,38,39)(27,41,43,40)(28,31,42,44);(2,4)(5,8,9,10)(6,7,11,12)(14,21)(15,19)(17,30)(20,23)(22,27)(24,37)(25,34)(26,32)(28,31)(33,36)(35,40)(38,43)(39,41)(42,44);(6,11)(7,12)(13,16,18,29)(14,22,24,38)(15,25,19,34)(17,31,32,44)(20,35,33,39)(21,27,37,43)(23,40,36,41)(26,42,30,28)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416501);