#6448752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437478
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3230 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)(3,6)(4,5)(7,8)(9,10,16,22)(11,23,25,35)(12,18,29,38)(13,19,31,40)(14,20,15,21)(17,32,36,24)(26,39,30,33)(27,34,28,37);(9,11)(10,17)(12,26)(13,28)(14,24)(15,32)(16,25)(18,19)(20,35)(21,23)(22,36)(27,31)(29,30)(33,37)(34,39)(38,40);(2,5)(6,8)(9,12)(10,18)(11,13)(14,28)(15,27)(16,29)(17,33)(19,23)(20,37)(21,34)(22,38)(24,30)(25,31)(26,32)(35,40)(36,39);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,12)(14,26)(15,30)(16,31)(17,34)(18,23)(20,39)(21,33)(22,40)(24,27)(25,29)(28,32)(35,38)(36,37);(9,14,16,15)(10,20,22,21)(11,24,25,32)(12,27,29,28)(13,26,31,30)(17,35,36,23)(18,34,38,37)(19,39,40,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355948);