#6452239

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
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3190 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)(9,10,15,21)(11,17,25,31)(12,18,28,36)(13,19,14,20)(16,23,32,24)(22,33,38,40)(26,39,37,30)(27,34,29,35);(9,11,14,23,15,25,13,24)(10,16,19,31,21,32,20,17)(12,26,29,38,28,37,27,22)(18,33,34,30,36,40,35,39);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,22)(13,24)(14,23)(15,25)(16,19)(18,33)(20,32)(21,31)(26,29)(27,37)(28,38)(30,34)(35,39)(36,40);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,22)(13,27)(14,29)(15,28)(16,30)(17,33)(19,34)(20,35)(21,36)(23,26)(24,37)(25,38)(31,40)(32,39);(9,13,15,14)(10,19,21,20)(11,23,25,24)(12,27,28,29)(16,17,32,31)(18,34,36,35)(22,26,38,37)(30,33,39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,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,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, 6359435);