#6573455

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

4560 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,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, 6480651);