#441784

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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

919 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efgi,c^2=ef,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 348980);