Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25271 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2126 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)(11,12,17,30,16,21,15,20)(13,18,31,42,24,34,23,33)(14,25,27,19,29,36,28,37)(22,38,39,32,26,41,40,35);(4,7)(8,10)(11,13,15,23,16,24,17,31)(12,18,20,33,21,34,30,42)(14,26,28,39,29,22,27,40)(19,35,25,41,37,32,36,38);(3,5,6,9)(4,8,7,10)(11,14,16,29)(12,19,21,37)(13,22,24,26)(15,27,17,28)(18,32,34,35)(20,36,30,25)(23,39,31,40)(33,41,42,38);(4,7)(8,10)(11,15,16,17)(12,20,21,30)(13,23,24,31)(14,27,29,28)(18,33,34,42)(19,36,37,25)(22,39,26,40)(32,41,35,38)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348891);