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×#50751 |
| 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 | |
5758 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);(11,12)(13,15)(14,18)(16,17)(19,20,25,27)(21,28,22,23)(24,32,33,26)(29,31,30,34);(3,4)(5,7)(6,9)(8,10)(11,13)(12,15)(14,16)(17,18)(19,21,25,22)(20,23,27,28)(24,29,33,30)(26,31,32,34);(4,5)(8,9)(12,17)(13,14)(19,22,25,21)(20,23,27,28)(24,30,33,29)(26,31,32,34);(3,5,7,4)(6,9,10,8)(11,14,16,13)(12,18,17,15)(19,23)(20,21)(22,27)(24,31)(25,28)(26,29)(30,32)(33,34);(3,6)(4,8)(5,9)(7,10)(11,15,16,18)(12,13,17,14)(19,24,25,33)(20,26,27,32)(21,29,22,30)(23,31,28,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613538);