#7564448

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

1828 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=g,e^2=f,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, 7471644);