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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,7)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,31,28)(27,34,29,32)(30,36,38,33)(35,39,37,40);(1,2)(3,4)(5,6)(7,8)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,26,31,28)(27,32,29,34)(30,33,38,36)(35,39,37,40);(1,3)(2,4)(5,7)(6,8)(10,15)(11,18)(16,21)(19,24)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(1,2)(3,8)(4,7)(5,6)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(1,4,5,8)(2,7,6,3)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(1,2)(3,4)(5,6)(7,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,30,31,38)(26,33,28,36)(27,35,29,37)(32,39,34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9613551);