#7562565

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	#60651	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2070 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,24)(13,30)(14,22)(15,20)(17,29)(18,31)(19,26)(21,27)(23,32)(25,33)(28,37)(34,39)(35,36)(38,40);(2,5)(6,7)(9,11,15,18)(10,16,20,31)(12,25,28,38)(13,19,32,22)(14,30,26,23)(17,27,35,39)(21,36,34,29)(24,33,37,40);(9,12,26,36,15,28,14,29)(10,17,22,37,20,35,19,24)(11,21,30,38,18,34,23,25)(13,27,16,33,32,39,31,40);(9,13,14,31,15,32,26,16)(10,18,19,30,20,11,22,23)(12,27,29,40,28,39,36,33)(17,34,24,38,35,21,37,25);(1,3,4,8)(2,6,5,7)(12,28)(17,35)(21,34)(24,37)(25,38)(27,39)(29,36)(33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=g,e^2=f,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,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, 7469761);