#7565441

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	#3498	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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)(11,21)(12,13)(14,19)(15,33)(16,24)(17,28)(18,31)(20,32)(22,39)(23,26)(25,36)(27,30)(29,40)(34,38)(35,37);(2,6)(7,8)(9,11,14,24)(10,16,19,21)(12,26,27,40)(13,29,30,23)(15,34,32,25)(17,37,31,39)(18,35,28,22)(20,38,33,36);(1,3,5,4)(2,7,6,8)(9,12,15,28,14,27,32,18)(10,17,20,13,19,31,33,30)(11,22,25,26,24,35,34,40)(16,29,36,37,21,23,38,39);(1,3,5,4)(2,7,6,8)(9,13,15,31,14,30,32,17)(10,18,20,12,19,28,33,27)(11,23,25,39,24,29,34,37)(16,35,36,40,21,22,38,26);(1,4,5,3)(2,7,6,8)(10,19)(12,27)(16,21)(18,28)(20,33)(22,35)(26,40)(36,38)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fgi,e^2=fi,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, 7472637);