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×#56608 |
| 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 | |
5442 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);(4,7)(8,10)(11,12,14,16)(13,17,18,15)(19,20,23,24)(21,27,28,25)(22,30,31,26)(29,32,34,33);(5,9)(8,10)(13,18)(15,17)(20,24)(21,28)(26,30)(29,34);(3,4,6,7)(5,8,9,10)(11,13)(12,15)(14,18)(16,17)(19,21,23,28)(20,25,24,27)(22,29,31,34)(26,32,30,33);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,34);(3,5)(4,8)(6,9)(7,10)(19,22)(20,26)(21,29)(23,31)(24,30)(25,32)(27,33)(28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006704);