Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9290 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,11)(6,14,16,9)(7,13)(8,10)(12,15,18,17)(19,20,23,25,24,28,22,27)(21,29,31,26,32,33,30,34);(3,5,11,4,8,13,7,10)(6,12,15,9,16,18,17,14)(20,25)(22,23)(26,29)(27,28)(30,31)(33,34);(3,6)(4,9)(5,12)(7,17)(8,16)(10,14)(11,15)(13,18)(19,21,24,32)(20,26,28,34)(22,31,23,30)(25,29,27,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106127);