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 | |
4348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(10,15)(13,17)(16,19)(21,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,17)(13,18)(14,15)(16,22)(20,21)(23,24)(26,30)(31,33)(32,36)(38,39);(10,15)(12,18)(16,19)(20,24)(25,26,29,30)(27,33,35,31)(28,32,37,36)(34,39,40,38);(9,11,14,22)(10,16,15,19)(12,20,18,24)(13,21,17,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(9,12,14,18)(10,17,15,13)(11,20,22,24)(16,23,19,21)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,3,4,7)(2,6,5,8)(9,13,14,17)(10,18,15,12)(11,21,22,23)(16,24,19,20)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|c^2=h,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622107);