#6629712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629034
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2120 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,18,14)(12,16,24,30)(13,26,28,17)(21,33,31,25)(22,27,32,34)(23,36,37,29)(35,38,39,40);(1,3,4,7)(2,5,6,8)(9,11,14,10,15,18,20,19)(12,21,25,16,24,31,33,30)(13,22,27,17,28,32,34,26)(23,35,38,29,37,39,40,36);(2,6)(5,8)(9,12)(10,16)(11,21)(13,23)(14,25)(15,24)(17,29)(18,31)(19,30)(20,33)(22,35)(26,36)(27,38)(28,37)(32,39)(34,40);(9,13,14,27,15,28,20,34)(10,17,18,32,19,26,11,22)(12,23,25,38,24,37,33,40)(16,29,31,39,30,36,21,35);(12,24)(16,30)(21,31)(23,37)(25,33)(29,36)(35,39)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536908);