Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,17)(11,18,21,16)(12,13,23,14)(19,22,24,20)(25,26,29,31)(27,33,35,30)(28,36,38,32)(34,39,37,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,19)(13,20)(14,22)(15,21)(17,18)(23,24)(25,27)(26,30)(28,37)(29,35)(31,33)(32,40)(34,38)(36,39);(2,6)(3,8)(9,12)(10,13)(11,19)(14,17)(15,23)(16,20)(18,22)(21,24)(26,31)(28,38)(30,33)(34,37);(2,6)(3,8)(9,13,15,14)(10,12,17,23)(11,20,21,22)(16,19,18,24)(25,28,29,38)(26,32,31,36)(27,34,35,37)(30,39,33,40);(2,6)(5,7)(11,21)(16,18)(19,24)(20,22)(26,31)(27,35)(32,36)(34,37);(9,14,15,13)(10,12,17,23)(11,22,21,20)(16,19,18,24)Polycyclic presentation:
2^9|a^2=gi,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011641);