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 | |
4464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(25,26)(27,29)(28,36)(30,32)(31,34)(33,37)(35,39)(38,40);(1,2,4,5)(3,6,7,8)(25,27,30,34)(26,31,32,29)(28,35,37,40)(33,39,36,38);(10,15)(11,18)(17,20)(19,24)(26,32)(27,34)(28,37)(38,39);(1,3,4,7)(2,6,5,8)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,37)(38,40);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,29,30,31)(26,34,32,27)(28,38,37,39)(33,40,36,35)Polycyclic presentation:
2^9|b^2=g,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618315);