#507991

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eg,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, 415187);