#7542057

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

1701 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)(3,6)(4,8)(5,7)(9,10,15,21)(11,22,12,17)(13,28,30,36)(14,20,26,19)(16,25,32,24)(18,34,37,29)(23,35,27,31)(33,39,38,40);(1,3,5,8)(2,6,7,4)(9,11,14,24,15,12,26,25)(10,16,19,22,21,32,20,17)(13,27,29,40,30,23,34,39)(18,35,28,33,37,31,36,38);(1,4,5,6)(2,3,7,8);(2,7)(3,8)(9,12)(10,17)(11,15)(13,27)(14,25)(16,19)(18,33)(20,32)(21,22)(23,30)(24,26)(28,31)(29,40)(34,39)(35,36)(37,38);(9,13,26,34,15,30,14,29)(10,18,20,36,21,37,19,28)(11,23,25,40,12,27,24,39)(16,31,17,33,32,35,22,38)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,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,e^b=ei,h^b=hi,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, 7449253);