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 | |
4547 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,13)(14,15)(16,21)(17,24)(19,23)(20,22)(25,26,30,32)(27,31,34,28)(29,37,39,33)(35,36,40,38);(1,2)(3,4)(5,6)(7,8)(10,15)(12,17)(16,18)(19,22)(25,27,30,34)(26,31,32,28)(29,35,39,40)(33,38,37,36);(2,6)(3,7)(10,15)(12,17)(16,18)(19,22)(26,32)(27,34)(33,37)(35,40);(3,7)(4,8)(9,11)(10,16)(12,22)(13,23)(14,21)(15,18)(17,19)(20,24)(25,28)(26,27)(29,38)(30,31)(32,34)(33,40)(35,37)(36,39);(1,3)(2,4)(5,7)(6,8)(9,12,14,17)(10,13,15,24)(11,19,21,22)(16,20,18,23)(25,27)(26,31)(28,32)(29,35)(30,34)(33,38)(36,37)(39,40);(1,4)(2,3)(5,8)(6,7)(9,13)(10,17)(11,20)(12,15)(14,24)(16,22)(18,19)(21,23)(25,29,30,39)(26,33,32,37)(27,35,34,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8972577);