#501827

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	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

699 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)(13,14,18,24)(15,21,19,33)(16,27,29,43)(17,20,32,36)(22,38,39,28)(23,26,35,30)(25,42,34,41)(31,37,44,40);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,20)(15,19)(17,30)(21,23)(22,27)(24,36)(25,34)(26,32)(28,31)(33,35)(37,41)(38,44)(39,43)(40,42);(2,4)(6,10)(9,12)(13,15,18,19)(14,21,24,33)(16,25,29,34)(17,26,32,30)(20,35,36,23)(22,37,39,40)(27,42,43,41)(28,31,38,44);(13,16,18,29)(14,22,24,39)(15,25,19,34)(17,31,32,44)(20,27,36,43)(21,37,33,40)(23,41,35,42)(26,38,30,28)

Polycyclic presentation:

2^9|a^2=i,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, 409023);