#503629

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

686 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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, 410825);