#6440374

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	C₂×#3489	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4206 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,24,31)(12,17,27,36)(13,28,14,18)(16,22,32,23)(21,37,39,33)(25,30,38,40)(26,35,29,34);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,25)(13,22)(14,23)(15,24)(17,33)(18,31)(19,32)(20,28)(21,26)(27,38)(29,39)(30,35)(34,40)(36,37);(2,6)(5,8)(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,34)(19,36)(20,37)(22,38)(23,25)(24,39)(28,35)(31,33)(32,40);(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,34)(19,36)(20,37)(22,38)(23,25)(24,39)(28,35)(31,33)(32,40);(9,13,15,14)(10,18,19,28)(11,22,24,23)(12,26,27,29)(16,31,32,20)(17,34,36,35)(21,38,39,25)(30,33,40,37)

Polycyclic presentation:

2^9|a^2=i,b^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6347570);