Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53288 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2682 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5168 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);(4,7)(5,6)(13,17)(16,18)(19,20)(21,25)(22,24)(23,30)(26,31)(27,28)(29,34)(32,33);(3,4,8,7)(5,10,6,9)(12,15)(16,18)(20,22)(21,27)(23,31)(32,34);(3,5,8,6)(4,9,7,10)(11,12)(13,16)(14,15)(17,18)(19,21,24,27)(20,25,22,28)(23,29,31,33)(26,32,30,34);(3,6,8,5)(4,10,7,9)(11,13)(12,16)(14,17)(15,18)(19,22,24,20)(21,28,27,25)(23,30,31,26)(29,34,33,32);(3,7,8,4)(5,10,6,9)(11,12)(13,16)(14,15)(17,18)(19,23)(20,26)(21,29)(22,30)(24,31)(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, 8012908);