#503290

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

748 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)(15,23)(16,24)(17,27)(19,29)(21,30)(22,34)(25,39)(26,37)(28,38)(31,40)(32,44)(33,43)(35,36)(41,42);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14)(15,19)(16,21)(17,22)(18,20)(23,29)(24,30)(25,32)(26,31)(27,34)(28,33)(35,41)(36,42)(37,40)(38,43)(39,44);(2,4)(13,15,24,36,17,25,28,37,18,26,38,39,27,35,16,23)(14,19,30,42,22,32,33,40,20,31,43,44,34,41,21,29);(5,8,9,7)(6,11,12,10)(14,20)(19,31)(21,33)(22,34)(29,40)(30,43)(32,41)(42,44)

Polycyclic presentation:

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

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, 410486);