Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61344 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6396 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);(4,7)(8,9)(11,12)(13,21)(14,25)(15,19)(16,29)(17,23)(18,27)(20,28)(22,37)(24,32)(26,34)(30,33)(31,41)(35,39)(36,40)(38,42);(3,4,6,7)(5,9,10,8)(11,13,15,23)(12,17,19,21)(14,22,26,38)(16,24,28,40)(18,31,33,35)(20,32,29,36)(25,42,34,37)(27,39,30,41);(11,14,15,26)(12,18,19,33)(13,22,23,38)(16,30,28,27)(17,31,21,35)(20,34,29,25)(24,41,40,39)(32,37,36,42);(3,5,6,10)(4,8,7,9)(13,23)(17,21)(22,38)(24,40)(31,35)(32,36)(37,42)(39,41);(12,19)(13,23)(18,33)(20,29)(22,38)(24,40)(25,34)(39,41)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483703);