#500876

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

775 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=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, 408072);