#500089

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
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	C₂⁵	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^c=fh,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, 407285);