Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#8877 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #385 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
912 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);(3,4,9,14)(5,15,6,17)(7,13,8,12)(10,18,11,16)(19,20,23,28,25,22,24,27)(21,29,32,30,33,34,31,26);(3,5,9,6)(4,10,14,11)(7,18,8,16)(12,17,13,15)(19,21,23,32,25,33,24,31)(20,26,28,29,22,30,27,34);(3,6,9,5)(4,11,14,10)(7,18,8,16)(12,17,13,15)(19,22,25,20)(21,30,33,26)(23,27,24,28)(29,32,34,31)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99687);