Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#7700 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,8,7,10,5,9)(11,12,15,17,16,13,14,18)(19,20)(21,27)(22,30)(23,25)(24,28)(26,32)(29,34)(31,33);(4,8)(5,6)(9,10)(11,13,16,12)(14,17,15,18)(19,21,23,29)(20,24,25,33)(22,31,32,28)(26,34,30,27);(3,5,7,6)(4,8,10,9)(12,13)(17,18)(20,25)(24,33)(26,30)(27,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 68813);