Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9450 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1130 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)(5,12,13,16)(6,9,14,15)(7,10)(8,11)(17,18)(19,20,23,28,24,27,22,26)(21,25,29,32,30,33,31,34);(3,5,7,15,8,13,17,9)(4,6,10,16,11,14,18,12)(19,21,22,29,24,30,23,31)(20,25,28,34,27,33,26,32);(3,6)(4,9)(5,10)(7,16)(8,14)(11,15)(12,17)(13,18)(19,22,24,23)(20,26,27,28)(21,29,30,31)(25,32,33,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108329);