#443004

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

775 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)(6,7)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,31)(21,23)(22,35)(24,30)(25,32)(26,33)(28,41)(29,39)(34,38)(36,43)(37,48)(40,42)(44,45)(46,47);(1,2,4,5)(3,7,8,6)(9,11,12,15)(10,13,14,16)(18,23)(20,32)(24,36)(26,35)(27,39)(28,42)(34,47)(44,48);(11,15)(13,16)(17,19,22,30,21,29,33,43)(18,24,26,27,23,36,35,39)(20,28,34,44,32,42,47,48)(25,37,38,40,31,45,46,41);(1,3,4,8)(2,6,5,7)(17,20,33,47,21,32,22,34)(18,25,35,46,23,31,26,38)(19,28,43,48,29,42,30,44)(24,37,39,41,36,45,27,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,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^b=de,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, 350200);