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×#49073 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2670 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,10)(7,8)(11,12,16,17)(13,18,15,14)(20,24)(25,28)(26,31)(32,34);(3,5)(4,6)(7,10)(8,9)(12,17)(13,15)(20,24)(21,27)(22,30)(32,34);(4,8)(5,10)(11,13,16,15)(12,18,17,14)(19,20,23,24)(21,28,27,25)(22,26,30,31)(29,34,33,32);(3,5)(4,9)(6,8)(7,10)(11,14,16,18)(12,13,17,15)(19,21,23,27)(20,25,24,28)(22,29,30,33)(26,32,31,34);(3,6,7,9)(4,5,8,10)(11,15,16,13)(12,18,17,14)(19,22)(20,26)(21,29)(23,30)(24,31)(25,32)(27,33)(28,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612005);