#6573912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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

2659 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)(9,10,16,15)(11,17,23,32)(12,18,14,20)(13,19,29,33)(21,30,36,24)(22,34,26,35)(25,27,31,28)(37,39,40,38);(1,2,3,4)(9,11,16,23)(10,17,15,32)(12,24,14,30)(13,26,29,22)(18,21,20,36)(19,35,33,34)(25,39,31,38)(27,40,28,37);(6,8)(9,12,16,14)(10,18,15,20)(11,21,23,36)(13,27,29,28)(17,30,32,24)(19,31,33,25)(22,38,26,39)(34,37,35,40);(9,13,16,29)(10,19,15,33)(11,22,23,26)(12,25,14,31)(17,34,32,35)(18,27,20,28)(21,37,36,40)(24,38,30,39);(9,14,16,12)(10,20,15,18)(11,21,23,36)(13,28,29,27)(17,30,32,24)(19,25,33,31)(22,38,26,39)(34,37,35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481108);