#7562788

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519075
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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)(9,10,13,15)(11,19,22,34)(12,24,25,39)(14,28,27,18)(16,20,29,23)(17,32,30,26)(21,36,38,31)(33,35,40,37);(2,5)(6,7)(10,15)(11,20)(12,25)(14,27)(16,19)(17,30)(21,37)(22,23)(29,34)(31,33)(35,38)(36,40);(9,11,14,23,13,22,27,20)(10,16,18,19,15,29,28,34)(12,21,26,37,25,38,32,35)(17,31,24,33,30,36,39,40);(9,12,14,26,13,25,27,32)(10,17,18,24,15,30,28,39)(11,21,23,37,22,38,20,35)(16,31,19,33,29,36,34,40);(1,3,4,8)(2,6,5,7)(10,15)(16,29)(17,30)(18,28)(19,34)(24,39)(31,36)(33,40)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=g,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=cgi,d^b=dh,g^b=gi,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, 7469984);