Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11309 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #387 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #1443 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,13,16,9)(6,10,18,25)(7,12)(8,11)(14,29)(15,30,33,23)(17,34)(19,26,20,28)(21,27)(22,31)(24,32);(3,5)(4,9)(6,17,18,32)(7,15)(8,16)(10,24,25,34)(11,13)(12,23)(14,19,31,20)(21,33)(22,26,29,28)(27,30);(3,6)(4,10,11,25)(5,14)(7,20)(8,18)(9,22,13,29)(12,28,27,26)(15,32)(16,31)(17,33)(19,21)(23,34,30,24)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265544);