#503292

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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

744 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)(15,24)(16,21)(17,32)(18,30)(19,23)(20,27)(22,25)(26,37)(28,43)(29,39)(31,36)(33,44)(34,41)(35,42)(38,40);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14)(15,20)(16,30)(17,22)(18,21)(19,23)(24,27)(25,32)(26,38)(28,34)(29,35)(31,44)(33,36)(37,40)(39,42)(41,43);(2,4)(13,15,25,40,18,28,33,42,19,29,31,41,16,26,17,27)(14,20,32,37,21,34,36,39,23,35,44,43,30,38,22,24);(5,8,9,7)(6,11,12,10)(13,16,19,18)(14,21,23,30)(15,26,29,28)(17,31,33,25)(20,34,35,38)(22,32,36,44)(24,37,39,43)(27,41,42,40)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=ei,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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410488);