Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51110 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(12,15)(16,18)(19,20,24,25)(21,22,26,29)(23,31,32,27)(28,34,33,30);(3,4,7,8)(5,10,9,6)(20,25)(22,29)(27,31)(30,34);(4,8)(5,9)(11,12)(13,16)(14,15)(17,18)(19,21,24,26)(20,22,25,29)(23,28,32,33)(27,30,31,34);(3,5,7,9)(4,6,8,10)(19,22)(20,26)(21,25)(23,30)(24,29)(27,33)(28,31)(32,34);(3,6)(4,9)(5,8)(7,10)(11,13,14,17)(12,16,15,18)(19,23)(20,27)(21,28)(22,30)(24,32)(25,31)(26,33)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614001);