#7564455

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

1820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,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, 7471651);