Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#56331 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2703 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5216 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);(11,12,14,16)(13,17,18,15)(19,20)(21,24)(22,27)(23,28)(25,29)(26,31)(30,32)(33,34);(3,4)(5,9)(6,8)(7,10)(13,18)(15,17)(20,21)(22,29)(26,28)(30,34);(3,4,6,8)(5,7,10,9)(11,13)(12,15)(14,18)(16,17)(19,21,24,20)(22,27,29,25)(23,28,31,26)(30,33,34,32);(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,25)(21,27)(23,30)(24,29)(26,32)(28,33)(31,34);(3,5)(4,7)(6,10)(8,9)(19,23,24,31)(20,26,21,28)(22,30,29,34)(25,32,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006144);