#410718

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

716 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)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,29)(20,33)(21,24)(22,39)(23,28)(25,36)(26,32)(27,30)(31,40)(34,42)(35,44)(37,43)(38,41)(45,46)(47,48);(2,5)(6,8)(11,15)(13,16)(17,19,30,38,23,32,22,21)(18,24,39,26,28,41,27,29)(20,31,36,46,37,47,35,34)(25,40,33,42,44,48,43,45);(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,31)(21,34)(22,35)(23,37)(24,40)(26,42)(27,43)(28,44)(29,45)(30,36)(32,47)(33,39)(38,46)(41,48);(1,3,4,7)(2,6,5,8)(17,21,22,32,23,38,30,19)(18,26,27,24,28,29,39,41)(20,34,35,47,37,46,36,31)(25,42,43,40,44,45,33,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317914);