#512478

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
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

426 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,c^b=ci,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419674);