Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
| direct product | C4×#2402 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C23 |
| commutator | C23 | C24×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,20)(17,19)(21,22,24,26)(23,25,27,28);(6,9)(10,12)(13,15,17,16)(14,18,19,20)(21,23)(22,25)(24,27)(26,28);(13,16,17,15)(14,18,19,20)(23,27)(25,28);(5,7,8,11)(6,10,9,12)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421239);