#466052

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

1053 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=ei,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,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, 373248);