#7533783

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

1941 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)(11,22,13,18)(12,17,27,20)(14,29,30,33)(16,25,32,23)(19,35,37,26)(24,36,28,31)(34,39,38,40);(3,8)(5,7)(9,11,12,23,15,13,27,25)(10,16,20,22,21,32,17,18)(14,28,26,40,30,24,35,39)(19,36,29,34,37,31,33,38);(1,3,4,8)(2,5,6,7)(9,12,15,27)(10,17,21,20)(11,23,13,25)(14,26,30,35)(16,18,32,22)(19,33,37,29)(24,39,28,40)(31,34,36,38);(2,6)(5,7)(9,13)(10,18)(11,15)(12,25)(14,28)(16,20)(17,32)(19,34)(21,22)(23,27)(24,30)(26,40)(29,31)(33,36)(35,39)(37,38);(9,14,12,26,15,30,27,35)(10,19,20,29,21,37,17,33)(11,24,23,39,13,28,25,40)(16,31,22,38,32,36,18,34)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,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,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, 7440979);