#6573748

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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

4515 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480944);