#6573921

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2623 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^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, 6481117);