#7560739

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
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	#61176	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1720 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)(11,16)(12,23)(13,21)(14,29)(15,19)(17,28)(18,25)(20,26)(22,34)(24,33)(27,38)(30,35)(31,32)(36,37)(39,40);(2,5)(6,8)(9,11,15,22)(10,16,19,34)(12,24,27,39)(13,29,31,18)(14,32,25,21)(17,26,36,30)(20,37,35,28)(23,33,38,40);(1,3,4,7)(2,6,5,8)(9,12,25,37,15,27,14,28)(10,17,29,38,19,36,18,23)(11,20,32,39,22,35,21,24)(13,30,34,40,31,26,16,33);(9,13,14,16,15,31,25,34)(10,11,18,21,19,22,29,32)(12,26,28,40,27,30,37,33)(17,35,23,39,36,20,38,24);(2,5)(6,8)(12,27)(17,36)(20,35)(23,38)(24,39)(26,30)(28,37)(33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=g,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,e^c=ei,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, 7467935);