#9937780

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9844976);