#7519257

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
prime radical	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2614 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)(9,10,14,16)(11,21,25,32)(12,26,28,34)(13,20,15,19)(17,24,30,22)(18,29,31,27)(23,33,38,39)(35,36,40,37);(1,2,4,3)(5,6)(7,8)(10,16)(11,22)(12,27)(13,15)(17,32)(18,34)(21,30)(24,25)(26,31)(28,29)(33,39)(36,37);(9,11,14,25)(10,17,16,30)(12,23,28,38)(13,22,15,24)(18,33,31,39)(19,21,20,32)(26,35,34,40)(27,36,29,37);(6,8)(9,12,14,28)(10,18,16,31)(11,23,25,38)(13,27,15,29)(17,33,30,39)(19,26,20,34)(21,35,32,40)(22,36,24,37);(1,3,4,2)(9,13,14,15)(10,19,16,20)(11,24,25,22)(12,27,28,29)(17,32,30,21)(18,26,31,34)(23,37,38,36)(33,40,39,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,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, 7426453);