#501835

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

720 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)(15,21)(16,27)(17,20)(18,24)(19,33)(22,29)(23,30)(25,42)(26,35)(28,38)(31,37)(32,36)(34,41)(39,43)(40,44);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,20)(15,19)(16,28)(17,30)(21,23)(22,38)(24,36)(26,32)(27,39)(29,44)(31,43)(33,35)(37,42)(40,41);(2,4)(6,10)(9,12)(13,15,18,19)(14,21,24,33)(16,25,28,34)(17,26,32,30)(20,35,36,23)(22,37,39,40)(27,42,38,41)(29,31,43,44);(13,16,18,28)(14,22,24,39)(15,25,19,34)(17,31,32,44)(20,27,36,38)(21,37,33,40)(23,41,35,42)(26,43,30,29)

Polycyclic presentation:

2^9|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,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, 409031);