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 | |
4370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,19)(12,15)(13,17)(14,18)(16,21)(20,22)(23,24);(3,8)(4,7)(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,23,22,18)(26,30)(31,37)(32,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,10,15,12)(11,19,21,16)(13,17,20,22)(14,18,24,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(1,4)(2,3)(5,7)(6,8)(9,12,15,10)(11,16,21,19)(13,22,20,17)(14,23,24,18);(9,13)(10,17)(11,14)(12,22)(15,20)(16,23)(18,19)(21,24)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,17)(19,22)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003333);