#508904

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,18,21)(12,16,20,22)(17,23,19,24)(25,26,29,36,44,53,43,33,31,34,28,35,30,37,45,52)(27,32,40,49,56,46,55,51,42,47,39,48,41,50,38,54);(1,3,4,7)(2,5,6,8)(10,14)(15,21)(16,22)(23,24)(25,27,28,39,44,56,45,38,31,42,29,40,30,41,43,55)(26,32,35,48,53,46,52,54,34,47,36,49,37,50,33,51);(2,6)(5,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(26,33)(27,38)(28,43)(29,45)(30,44)(32,46)(34,52)(35,37)(36,53)(39,56)(40,41)(42,55)(47,50)(51,54);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(26,34)(32,47)(33,52)(35,36)(37,53)(46,50)(48,49)(51,54)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=e,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fh,g^c=gh,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, 416100);