#460851

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368047);