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 | |
4290 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)(11,18)(12,13)(14,15)(16,19)(17,22)(20,24)(21,23)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40);(2,5)(3,7)(10,15)(12,17)(16,18)(20,23)(26,28)(31,33)(32,36)(38,39);(1,2)(3,6)(4,5)(7,8)(11,19)(16,18)(20,23)(21,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,18)(17,23)(22,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(9,12,14,17)(10,13,15,22)(11,20,19,23)(16,21,18,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39);(9,13)(10,17)(11,21)(12,15)(14,22)(16,23)(18,20)(19,24)Polycyclic presentation:
2^9|c^2=hi,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000676);