#6445670

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 6352866);