#500095

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
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×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1002 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)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,27,38)(16,22,18,34)(17,30,33,44)(21,39,41,26)(23,29,42,32)(25,35,28,37)(31,36,43,40);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,28)(17,26)(18,25)(19,27)(21,30)(22,37)(23,36)(24,38)(29,43)(31,32)(33,39)(34,35)(40,42)(41,44);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,21)(16,25)(17,31)(18,28)(19,27)(20,30)(22,36)(23,37)(24,41)(26,32)(29,39)(33,43)(34,40)(35,42)(38,44);(6,9)(10,12)(13,16,19,18)(14,22,24,34)(15,25,27,28)(17,29,33,32)(20,35,38,37)(21,36,41,40)(23,30,42,44)(26,31,39,43)

Polycyclic presentation:

2^9|a^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,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407291);