#7597055

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563327
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

2014 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)(19,20)(21,31)(24,40)(26,27)(34,38)(35,37);(1,2)(3,6)(4,5)(7,8)(9,10,14,17,16,21,15,31)(11,23,26,33,27,39,22,18)(12,29,13,20,30,32,28,19)(24,35,25,38,40,34,36,37);(2,5)(6,7)(9,11,15,22,16,27,14,26)(10,18,17,23,21,33,31,39)(12,24,28,36,30,40,13,25)(19,34,29,37,20,35,32,38);(1,3,4,8)(2,6,5,7)(9,12,15,28,16,30,14,13)(10,19,17,29,21,20,31,32)(11,24,22,36,27,40,26,25)(18,34,23,37,33,35,39,38);(9,13,15,12,16,28,14,30)(10,20,31,29,21,19,17,32)(11,25,22,24,27,36,26,40)(18,35,39,37,33,34,23,38)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=g,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^d=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, 7504251);