#7560725

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519075
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=gi,g^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,d^c=di,h^d=hi

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, 7467921);