#500121

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	C₂×#3285	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1025 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=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, 407317);