#512503

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=ef,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,g^a=gi,h^a=hi,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, 419699);