#6573926

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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

2599 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)(12,23)(17,24)(21,29)(25,26)(27,32)(34,40)(36,39)(37,38);(1,2,3,4)(9,10,15,19)(11,20,13,28)(12,24,23,17)(14,31,33,18)(16,30,35,22)(21,39,29,36)(25,40,26,34)(27,37,32,38);(6,8)(9,11,15,13)(10,16,19,35)(12,25,23,26)(14,30,33,22)(17,36,24,39)(18,28,31,20)(21,27,29,32)(34,37,40,38);(9,12,15,23)(10,17,19,24)(11,21,13,29)(14,32,33,27)(16,34,35,40)(18,38,31,37)(20,36,28,39)(22,25,30,26);(9,13,15,11)(10,16,19,35)(12,26,23,25)(14,22,33,30)(17,36,24,39)(18,28,31,20)(21,32,29,27)(34,37,40,38)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481122);