Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #40531 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(9,10)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26,31,29)(27,35,37,32)(28,38,36,33)(30,34,40,39);(3,4)(7,8)(10,15)(12,18)(17,21)(20,22)(25,27,31,37)(26,32,29,35)(28,30,36,40)(33,34,38,39);(1,3,6,4)(2,7,5,8)(10,15)(11,16)(17,21)(19,24);(3,4)(7,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,22)(23,24)(25,28,31,36)(26,33,29,38)(27,30,37,40)(32,34,35,39);(1,4,6,3)(2,7,5,8)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,29,31,26)(27,35,37,32)(28,33,36,38)(30,34,40,39);(1,5,6,2)(3,8,4,7)(9,13,14,23)(10,17,15,21)(11,19,16,24)(12,20,18,22)(25,30)(26,34)(27,36)(28,37)(29,39)(31,40)(32,38)(33,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050753);