#6573462

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

4541 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)(9,10,15,12)(11,16,23,24)(13,17,30,25)(14,18,31,26)(19,21,33,22)(20,32,37,39)(27,29,34,28)(35,36,40,38);(2,4)(9,11)(10,16)(12,24)(13,20)(14,21)(15,23)(17,32)(18,33)(19,26)(22,31)(25,39)(27,40)(28,36)(29,38)(30,37)(34,35);(6,8)(9,12,15,10)(11,19,23,33)(13,27,30,34)(14,18,31,26)(16,21,24,22)(17,29,25,28)(20,32,37,39)(35,38,40,36);(9,13,15,30)(10,17,12,25)(11,20,23,37)(14,28,31,29)(16,32,24,39)(18,27,26,34)(19,35,33,40)(21,36,22,38);(11,21)(13,28)(14,31)(16,33)(17,27)(18,26)(19,24)(22,23)(25,34)(29,30)(35,40)(36,38)

Polycyclic presentation:

2^9|a^2=i,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, 6480658);