#507808

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

454 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)(3,7)(4,6)(5,8)(17,18,22,28,21,27,36,43,23,29,38,46,24,30,39,40)(19,25,32,42,34,45,37,26,20,33,35,47,31,41,48,44);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,23)(21,34)(22,32)(24,31)(26,43)(27,45)(28,42)(29,33)(30,41)(35,38)(36,37)(39,48)(40,44)(46,47);(10,13)(14,16)(17,20,23,19)(18,26,29,44)(21,34,24,31)(22,37,38,48)(25,40,33,43)(27,42,30,47)(28,45,46,41)(32,39,35,36);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,27,29,30)(19,31,20,34)(22,36,38,39)(25,41,33,45)(26,42,44,47)(28,43,46,40)(32,48,35,37)

Polycyclic presentation:

2^9|a^2=f,b^2=e,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, 415004);