#7527626

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1903 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,e^2=f,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,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, 7434822);