#7533794

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

1900 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)(4,6)(5,8)(9,10,15,20)(11,17,23,31)(12,24,25,38)(13,28,14,19)(16,29,32,22)(18,27,34,26)(21,35,36,30)(33,39,40,37);(3,8)(5,7)(9,11,14,29,15,23,13,22)(10,16,28,31,20,32,19,17)(12,21,27,39,25,36,26,37)(18,30,38,40,34,35,24,33);(1,3,4,8)(2,5,6,7)(12,25)(18,34)(21,36)(24,38)(26,27)(30,35)(33,40)(37,39);(2,6)(5,7)(9,11)(10,17)(12,21)(13,22)(14,29)(15,23)(16,28)(18,33)(19,32)(20,31)(24,35)(25,36)(26,37)(27,39)(30,38)(34,40);(9,12,14,27,15,25,13,26)(10,18,28,38,20,34,19,24)(11,21,29,39,23,36,22,37)(16,30,31,40,32,35,17,33)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,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, 7440990);