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×#54024 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2617 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5426 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)(5,9)(12,15)(13,17)(19,20,23,26)(21,27,29,24)(22,25,30,32)(28,33,34,31);(3,4,6,7)(5,8,9,10)(12,15)(16,18)(22,30)(25,32)(28,34)(31,33);(3,4)(5,8)(6,7)(9,10)(11,12,14,15)(13,16,17,18)(19,21,23,29)(20,24,26,27)(22,28,30,34)(25,31,32,33);(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,17)(15,18)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,32)(31,34);(3,5)(4,8)(6,9)(7,10)(19,22)(20,25)(21,28)(23,30)(24,31)(26,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008485);