#6435910

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433049
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=f,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di

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, 6343106);