#507757

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
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

453 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)(17,18,22,28)(19,31,20,25)(21,30,24,27)(23,38,37,41)(26,33,45,35)(29,36,47,40)(32,42,34,46)(39,43,48,44);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,20)(18,25,28,31)(21,34,24,32)(23,33,37,35)(26,41,45,38)(27,46,30,42)(29,43,47,44)(36,39,40,48);(10,13)(14,16)(17,20,22,19)(18,26,28,45)(21,34,24,32)(23,39,37,48)(25,41,31,38)(27,44,30,43)(29,42,47,46)(33,40,35,36);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,24)(18,27,28,30)(19,32,20,34)(23,36,37,40)(25,42,31,46)(26,44,45,43)(29,41,47,38)(33,48,35,39)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=gi,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, 414953);