Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#23274 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,7)(8,10)(11,12)(13,16)(14,15)(17,18)(19,20,23,27,24,21,25,29)(22,26,31,33,30,28,32,34);(3,4,6,7)(5,8,9,10)(19,21,23,29,24,20,25,27)(22,28,31,34,30,26,32,33);(3,5,6,9)(4,8,7,10)(12,15)(16,18)(19,22)(20,26)(21,28)(23,31)(24,30)(25,32)(27,33)(29,34);(4,7)(8,10)(11,13,14,17)(12,16,15,18)(22,30)(26,28)(31,32)(33,34)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290482);