#7597045

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504241);