#6448662

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437473
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
prime radical	#62815	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3241 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,25)(14,17)(18,28)(19,29)(20,35)(21,30)(22,39)(23,33)(24,36)(26,31)(27,34)(32,38)(37,40);(9,11,14,21)(10,15,17,30)(12,22,23,38)(13,26,28,20)(16,32,33,39)(18,35,25,31)(19,27,37,24)(29,36,40,34);(2,5)(6,8)(9,12)(10,16)(11,19)(13,27)(14,23)(15,29)(17,33)(18,36)(20,38)(21,37)(22,26)(24,28)(25,34)(30,40)(31,39)(32,35);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,27)(14,23)(15,29)(17,33)(18,36)(20,38)(21,37)(22,26)(24,28)(25,34)(30,40)(31,39)(32,35);(10,17)(15,30)(16,33)(18,25)(29,40)(31,35)(32,39)(34,36)

Polycyclic presentation:

2^9|b^2=i,d^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,h^c=hi,f^d=fi,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, 6355858);