#6440610

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60369	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2731 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)(9,10,15,21,16,22,14,20)(11,23,26,35,27,36,32,17)(12,18,29,34,25,38,30,37)(13,19,28,33,24,40,31,39);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,28)(13,29)(14,26)(15,32)(16,27)(18,19)(20,23)(21,36)(22,35)(24,30)(25,31)(33,37)(34,39)(38,40);(2,6)(5,8)(9,12)(10,18)(11,24)(13,27)(14,30)(15,29)(16,25)(17,33)(19,36)(20,37)(21,34)(22,38)(23,40)(26,31)(28,32)(35,39);(9,13)(10,19)(11,25)(12,27)(14,31)(15,28)(16,24)(17,34)(18,36)(20,39)(21,33)(22,40)(23,38)(26,30)(29,32)(35,37);(9,14,16,15)(10,20,22,21)(11,26,27,32)(12,29,25,30)(13,31,24,28)(17,23,35,36)(18,34,38,37)(19,39,40,33)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,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, 6347806);