#6434419

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1348 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=fg,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,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341615);