Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×#1684 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C23 | C2×#121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,9,12)(7,13,16,20)(8,17,14,11)(10,18,19,15)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,36)(34,35);(5,7,14,18,9,16,8,15)(6,10,17,20,12,19,11,13)(22,26)(27,35)(28,33)(29,36);(21,23,24,30)(22,27,26,35)(25,34,32,31)(28,36,33,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 118331);