#6440357

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437445
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×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4162 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)(11,21)(12,17)(13,18)(14,28)(15,20)(16,25)(19,30)(22,37)(23,38)(24,36)(26,35)(27,34)(29,33)(31,40)(32,39);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,32)(18,31)(19,36)(20,33)(21,28)(25,30)(26,40)(27,39)(34,38)(35,37);(2,6)(5,8)(9,12,15,13)(10,17,20,18)(11,22,24,23)(14,27,30,26)(16,31,33,32)(19,35,28,34)(21,37,36,38)(25,40,29,39);(9,13,15,12)(10,18,20,17)(11,23,24,22)(14,26,30,27)(16,32,33,31)(19,34,28,35)(21,38,36,37)(25,39,29,40);(11,24)(16,33)(21,36)(22,23)(25,29)(31,32)(37,38)(39,40)

Polycyclic presentation:

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

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, 6347553);