#6573908

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

2639 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)(11,17)(12,18)(13,19)(14,16)(15,20)(21,34)(22,33)(23,30)(24,31)(25,35)(26,36)(27,37)(28,32)(29,38)(39,40);(1,2,3,4)(10,16)(11,13)(12,24)(15,22)(17,32)(19,30)(21,39)(23,28)(25,26)(34,35)(36,40)(37,38);(6,8)(9,11)(10,17)(12,25)(13,22)(14,23)(15,28)(16,30)(18,35)(19,33)(20,32)(21,29)(24,39)(26,27)(31,40)(34,38)(36,37);(9,12,14,24)(10,18,16,31)(11,21,23,26)(13,25,28,39)(15,29,22,27)(17,34,30,36)(19,35,32,40)(20,38,33,37);(9,13,14,28)(10,19,16,32)(11,22,23,15)(12,26,24,21)(17,33,30,20)(18,36,31,34)(25,27,39,29)(35,37,40,38)

Polycyclic presentation:

2^9|b^2=fi,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, 6481104);