#463283

order	512
nilpotency class	3
isoclinism	isoclinic class of #456400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

890 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ei,c^2=f,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,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, 370479);