#7519180

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

2668 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=fi,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,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, 7426376);