Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#45651 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2669 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(5,6)(8,9)(13,17)(16,18)(19,20)(21,25)(22,26)(23,24);(3,4,7,10)(5,9,6,8)(12,15)(13,17)(20,23)(21,26);(3,5,7,6)(4,8,10,9)(11,12)(13,16)(14,15)(17,18)(19,21,24,26)(20,25,23,22);(3,6,7,5)(4,9,10,8)(11,13)(12,16)(14,17)(15,18)(19,22)(20,21)(23,26)(24,25);(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,18)(19,23,24,20)(21,22,26,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609226);