#504134

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, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fi,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, 411330);