#7518578

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
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

1918 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)(11,21)(12,25)(13,20)(14,16)(15,19)(17,22)(18,26)(23,33)(24,32)(27,34)(28,30)(29,31)(35,37)(36,40)(38,39);(2,5)(6,7)(10,16)(11,22)(13,15)(17,32)(21,30)(23,37)(24,28)(25,34)(26,29)(33,40)(35,39)(36,38);(9,11,14,24)(10,17,16,30)(12,23,27,38)(13,22,15,28)(18,33,31,39)(19,21,20,32)(25,35,34,40)(26,36,29,37);(9,12,14,27)(10,18,16,31)(11,23,24,38)(13,26,15,29)(17,33,30,39)(19,25,20,34)(21,35,32,40)(22,36,28,37);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,19,16,20)(11,22,24,28)(12,26,27,29)(17,21,30,32)(18,25,31,34)(23,36,38,37)(33,35,39,40)

Polycyclic presentation:

2^9|c^2=i,d^2=i,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=cg,d^b=dh,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi

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, 7425774);