Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11550 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1088 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);(3,4,6,9,7,10,13,16)(5,8,12,15,11,14,17,18)(19,20)(21,25)(22,26,30,24)(23,27)(28,32,34,31)(29,33);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18)(20,24)(22,30)(25,31)(26,27)(28,34)(32,33);(5,11)(8,14)(12,17)(15,18)(19,21,23,29)(20,25,27,33)(22,28,30,34)(24,31,26,32)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387664);