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×#61552 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3515 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(4,7)(8,10)(11,12,17,21)(13,18,25,35)(14,27,30,38)(15,22,16,20)(19,31,37,29)(23,39,28,42)(24,34,26,33)(32,41,36,40);(3,4)(5,8)(6,7)(9,10)(11,13)(12,18)(14,28)(15,24)(16,26)(17,25)(19,36)(20,33)(21,35)(22,34)(23,30)(27,42)(29,41)(31,40)(32,37)(38,39);(11,14)(12,19)(13,23)(15,29)(16,31)(17,30)(18,32)(20,27)(21,37)(22,38)(24,40)(25,28)(26,41)(33,39)(34,42)(35,36);(4,7)(8,10)(11,15,17,16)(12,20,21,22)(13,24,25,26)(14,29,30,31)(18,33,35,34)(19,27,37,38)(23,40,28,41)(32,39,36,42);(3,5,6,9)(4,8,7,10)(12,21)(13,25)(19,37)(20,22)(23,28)(24,26)(27,38)(40,41)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484069);