#7533792

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

1946 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)(9,10,15,21)(11,22,12,17)(13,28,29,36)(14,20,26,19)(16,25,32,24)(18,34,35,30)(23,38,27,31)(33,39,37,40);(3,8)(5,7)(9,11,14,24,15,12,26,25)(10,16,19,22,21,32,20,17)(13,23,30,39,29,27,34,40)(18,31,28,37,35,38,36,33);(1,3,4,8)(2,5,6,7)(13,29)(18,35)(23,27)(28,36)(30,34)(31,38)(33,37)(39,40);(2,6)(5,7)(9,12)(10,17)(11,15)(13,27)(14,25)(16,19)(18,33)(20,32)(21,22)(23,29)(24,26)(28,31)(30,40)(34,39)(35,37)(36,38);(9,13,26,34,15,29,14,30)(10,18,20,36,21,35,19,28)(11,23,25,40,12,27,24,39)(16,31,17,33,32,38,22,37)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 7440988);