#7560724

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
prime radical	#61169	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 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)(13,22)(14,19)(15,30)(16,20)(18,28)(21,26)(23,32)(25,35)(27,37)(29,38)(31,36)(33,34)(39,40);(2,5)(6,8)(9,11,16,23)(10,17,20,32)(12,25,29,39)(13,30,33,19)(14,22,15,34)(18,26,37,31)(21,27,36,28)(24,35,38,40);(1,3,4,7)(2,6,5,8)(9,12,14,27,16,29,15,28)(10,18,30,38,20,37,19,24)(11,21,34,39,23,36,22,25)(13,31,32,40,33,26,17,35);(9,13,15,17,16,33,14,32)(10,11,19,22,20,23,30,34)(12,26,28,40,29,31,27,35)(18,36,24,39,37,21,38,25);(2,5)(6,8)(9,14,16,15)(10,19,20,30)(11,22,23,34)(12,27,29,28)(13,32,33,17)(18,24,37,38)(21,25,36,39)(26,35,31,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=g,e^2=i,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, 7467920);