Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11266 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #1442 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1574 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);(3,4,8,12,9,14,7,13)(5,15,17,10,19,26,18,27)(6,20,22,30,23,29,24,11)(16,32,33,28,34,25,21,31);(3,5,7,18,9,19,8,17)(4,10,13,15,14,27,12,26)(6,21,22,16,23,33,24,34)(11,28,20,25,30,31,29,32);(3,6,7,22,9,23,8,24)(4,11,12,29,14,30,13,20)(5,16,17,21,19,34,18,33)(10,25,15,31,27,32,26,28)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265476);