#441663

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
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

910 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,31)(21,36)(22,24)(23,29)(25,35)(26,33)(28,40)(30,42)(32,39)(34,41)(37,38)(43,46)(44,45)(47,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,30)(18,23,24,38)(20,32,35,47)(21,37,33,29)(25,40,31,45)(26,42,36,27)(28,43,44,34)(39,46,48,41);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(18,24)(23,38)(25,31)(26,36)(27,42)(39,48)(40,45)(41,46);(17,20,33,43,22,35,21,34)(18,25,36,46,24,31,26,41)(19,28,37,47,30,44,29,32)(23,39,42,45,38,48,27,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348859);