Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#20828 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,16,20,17,21,15,30)(13,18,25,33,23,32,24,39)(14,26,28,34,29,40,27,19)(22,35,38,41,36,42,37,31);(4,7)(8,10)(11,13)(12,18)(14,22)(15,24)(16,25)(17,23)(19,31)(20,33)(21,32)(26,35)(27,37)(28,38)(29,36)(30,39)(34,41)(40,42);(11,14,15,27,17,29,16,28)(12,19,20,26,21,34,30,40)(13,22,24,37,23,36,25,38)(18,31,33,35,32,41,39,42);(3,5,6,9)(4,8,7,10)(13,23)(18,32)(22,36)(24,25)(31,41)(33,39)(35,42)(37,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278392);