#7564202

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	#3512	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1647 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)(4,8)(5,6)(9,10)(11,16)(12,23)(13,21)(14,28)(15,19)(17,27)(18,31)(20,34)(22,33)(24,32)(25,37)(26,38)(29,30)(35,36)(39,40);(2,6)(7,8)(9,11,15,22)(10,16,19,33)(12,24,26,39)(13,28,29,18)(14,30,31,21)(17,34,35,25)(20,36,37,27)(23,32,38,40);(1,3,5,4)(2,7,6,8)(9,12,14,27,15,26,31,36)(10,17,18,23,19,35,28,38)(11,20,21,24,22,37,30,39)(13,25,16,32,29,34,33,40);(9,13,14,16,15,29,31,33)(10,11,18,21,19,22,28,30)(12,25,27,32,26,34,36,40)(17,20,23,24,35,37,38,39);(1,4,5,3)(2,7,6,8)(12,26)(17,35)(20,37)(23,38)(24,39)(25,34)(27,36)(32,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,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, 7471398);