#7562780

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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	#3035	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1635 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:

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

Polycyclic presentation:

2^9|a^2=fi,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,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469976);