Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60524 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,16,21)(13,17,24,31)(14,19,25,33)(15,18,27,20)(22,29,35,40)(23,28,37,30)(26,32,38,34)(36,39,42,41);(4,7)(8,10)(11,13,16,24)(12,17,21,31)(14,22,25,35)(15,23,27,37)(18,28,20,30)(19,29,33,40)(26,36,38,42)(32,39,34,41);(3,5,6,9)(4,8,7,10)(12,18)(15,27)(17,28)(19,32)(20,21)(23,37)(26,38)(29,39)(30,31)(33,34)(36,42)(40,41);(11,14)(12,19)(13,22)(15,26)(16,25)(17,29)(18,32)(20,34)(21,33)(23,36)(24,35)(27,38)(28,39)(30,41)(31,40)(37,42);(14,25)(19,33)(22,35)(26,38)(29,40)(32,34)(36,42)(39,41)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482233);