#6452234

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
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 ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3273 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)(3,8)(4,6)(5,7)(9,10)(11,22)(12,17)(13,18)(14,20)(15,21)(16,24)(19,26)(23,38)(25,33)(27,34)(28,32)(29,37)(30,35)(31,36)(39,40);(9,11,14,24,15,12,26,25)(10,16,19,22,21,33,20,17)(13,28,29,27,31,39,30,23)(18,34,35,32,36,38,37,40);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,15)(13,27)(14,25)(16,19)(18,34)(20,33)(21,22)(23,31)(24,26)(28,30)(29,39)(32,35)(36,38)(37,40);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,23)(12,27)(14,29)(15,31)(16,32)(17,34)(19,35)(20,37)(21,36)(22,38)(24,28)(25,39)(26,30)(33,40);(11,12)(16,33)(17,22)(23,27)(24,25)(28,39)(32,40)(34,38)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,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, 6359430);