#7597056

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563327
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

1960 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)(4,5)(6,8)(9,10,17,22)(11,23,27,38)(12,15,21,16)(13,29,32,20)(14,28,31,19)(18,33,36,26)(24,35,25,34)(30,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,26,17,27,16,33)(10,18,12,23,22,36,21,38)(13,30,14,25,32,40,31,24)(19,37,20,35,28,39,29,34);(2,5)(6,7)(9,12,16,10,17,21,15,22)(11,18,26,23,27,36,33,38)(13,20,31,19,32,29,14,28)(24,34,30,37,25,35,40,39);(1,3,4,8)(2,6,5,7)(9,13,16,31,17,32,15,14)(10,19,21,29,22,28,12,20)(11,24,26,30,27,25,33,40)(18,34,23,37,36,35,38,39);(9,14,16,13,17,31,15,32)(10,20,21,19,22,29,12,28)(11,25,33,30,27,24,26,40)(18,35,38,37,36,34,23,39)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=g,e^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

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, 7504252);