Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#8539 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,15,19,16,20,26,32)(13,17,23,29,22,28,36,40)(14,18,25,31,24,30,37,41)(21,33,35,42,34,27,38,39);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,21)(15,23)(16,22)(18,27)(19,29)(20,28)(24,34)(25,35)(26,36)(30,33)(31,39)(32,40)(37,38)(41,42);(11,14)(12,18)(13,21)(15,25)(16,24)(17,27)(19,31)(20,30)(22,34)(23,35)(26,37)(28,33)(29,39)(32,41)(36,38)(40,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95197);