#6573903

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

2695 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^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, 6481099);