#500907

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

780 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)(5,6,9,12)(7,11,8,10)(13,14,16,19)(15,23,25,37)(17,29,27,40)(18,32,28,22)(20,36,34,24)(21,39,35,30)(26,43,33,41)(31,42,44,38);(5,7,9,8)(6,10,12,11)(14,19)(15,24)(17,30)(18,28)(20,37)(21,40)(23,34)(25,36)(26,44)(27,39)(29,35)(31,33)(38,41)(42,43);(2,4)(13,15,16,25)(14,20,19,34)(17,31,27,44)(18,33,28,26)(21,41,35,43)(22,42,32,38)(23,40,37,29)(24,39,36,30);(5,8,9,7)(6,11,12,10)(14,19)(20,34)(21,35)(22,32)(23,37)(29,40)(38,42)(41,43)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=i,d^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, 408103);