Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C8 |
| direct product | C2×#8270 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C4×C4⋊C4 |
| prime radical | C24 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
818 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);(4,8,10,13)(9,15,16,17)(19,20)(21,24)(22,26)(23,25)(27,31)(28,30)(29,32)(33,34);(3,4)(5,9)(6,13)(7,10)(8,14)(11,17)(12,16)(15,18)(19,21,22,27,23,28,29,33)(20,24,26,31,25,30,32,34);(3,5,6,11,7,12,14,18)(4,9,13,17,10,16,8,15)(20,25)(24,30)(26,32)(31,34)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90534);