#411514

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

928 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,c^2=e,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi

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, 318710);