#7518991

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2629 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=i,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^c=ei

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, 7426187);