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×#7726 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,6,8,7,10,5,9)(11,12)(13,19,21,16)(14,22)(15,18)(17,23)(20,24,26,25)(27,28)(29,35,37,42)(30,34)(31,33)(32,36,40,38)(39,41);(4,8)(5,6)(9,10)(11,13)(12,16)(14,20)(15,21)(17,24)(18,19)(22,25)(23,26)(27,29)(28,32)(30,38)(31,37)(33,40)(34,42)(35,41)(36,39);(3,5,7,6)(4,8,10,9)(11,14,15,23)(12,17,18,22)(13,20,21,26)(16,24,19,25)(28,33)(32,40)(34,41)(35,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 68893);