#6438681

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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

3160 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)(11,16,23,31)(12,17,26,35)(13,27,14,18)(20,29,37,40)(21,36,22,30)(24,32,38,39)(25,34,28,33);(5,6,8,7)(9,11,15,23)(10,16,19,31)(12,24,26,38)(13,22,14,21)(17,32,35,39)(18,36,27,30)(20,28,37,25)(29,33,40,34);(2,4)(5,7,8,6)(9,12)(10,17)(11,20)(13,25)(14,28)(15,26)(16,29)(18,33)(19,35)(21,24)(22,38)(23,37)(27,34)(30,39)(31,40)(32,36);(9,12)(10,17)(11,20)(13,25)(14,28)(15,26)(16,29)(18,33)(19,35)(21,24)(22,38)(23,37)(27,34)(30,39)(31,40)(32,36);(9,13,15,14)(10,18,19,27)(11,21,23,22)(12,25,26,28)(16,30,31,36)(17,33,35,34)(20,24,37,38)(29,39,40,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345877);