#6573524

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4560 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)(3,4)(5,6)(7,8)(10,15)(12,23)(16,31)(18,29)(19,33)(20,37)(25,39)(26,27);(2,4)(9,10,13,15)(11,19,21,33)(12,24,23,17)(14,29,28,18)(16,30,31,22)(20,38,37,34)(25,40,39,32)(26,36,27,35);(6,8)(9,11,13,21)(10,16,15,31)(12,25,23,39)(14,30,28,22)(17,34,24,38)(18,33,29,19)(20,27,37,26)(32,35,40,36);(9,12)(10,17)(11,20)(13,23)(14,26)(15,24)(16,32)(18,36)(19,34)(21,37)(22,25)(27,28)(29,35)(30,39)(31,40)(33,38);(10,18)(12,26)(14,28)(15,29)(16,33)(17,24)(19,31)(20,25)(22,30)(23,27)(34,38)(37,39)

Polycyclic presentation:

2^9|b^2=i,c^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6480720);