Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7693 |
| 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 | |
978 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);(4,8,9,10)(11,12,15,17,16,13,14,18)(19,20)(21,27,29,24)(22,30)(23,26)(25,31)(28,32,34,33);(3,4)(5,8)(6,10)(7,9)(11,13,16,12)(14,17,15,18)(19,21,23,29)(20,24,26,27)(22,28,31,34)(25,32,30,33);(3,5,7,6)(4,8,9,10)(11,14,16,15)(12,17,13,18)(19,22,23,31)(20,25,26,30)(21,28,29,34)(24,32,27,33)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 68618);