#463059

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

777 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)(20,32)(21,23)(22,36)(24,31)(25,34)(26,28)(29,45)(30,41)(33,42)(35,40)(37,43)(38,48)(39,46)(44,47);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(24,37)(25,32)(26,36)(27,41)(38,39)(40,47)(42,45);(1,3)(2,6)(4,7)(5,8)(11,15)(13,16)(17,19,28,43,21,30,22,31)(18,24,36,41,23,37,26,27)(20,33,44,46,34,29,35,48)(25,39,47,42,32,38,40,45);(3,7)(6,8)(17,20,22,35,21,34,28,44)(18,25,26,40,23,32,36,47)(19,29,31,46,30,33,43,48)(24,38,27,42,37,39,41,45)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fi,d^2=f,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, 370255);