Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36769 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(5,9)(8,10)(13,17)(15,18)(19,20)(21,22)(23,27)(24,25)(26,30)(28,29)(31,32)(33,34);(4,7)(8,10)(11,12,14,16)(13,15,17,18)(20,25)(21,26)(27,32)(28,33);(3,4)(5,8)(6,7)(9,10)(11,13,14,17)(12,15,16,18)(19,21,24,26)(20,22,25,30)(23,28,31,33)(27,29,32,34);(3,4)(5,8)(6,7)(9,10)(19,22)(20,26)(21,25)(23,29)(24,30)(27,33)(28,32)(31,34);(3,5)(4,8)(6,9)(7,10)(19,23,24,31)(20,27,25,32)(21,28,26,33)(22,29,30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541380);