#6438663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,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, 6345859);