#7519245

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

2622 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426441);