#507993

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=egi,c^2=i,d^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415189);