Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61674 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7568 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)(5,9)(6,8)(7,10)(11,12,17,22)(13,18,26,36)(14,20,28,38)(15,21,16,31)(19,30,39,27)(23,40,24,33)(25,34,29,35)(32,41,37,42);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,24)(15,29)(16,25)(17,26)(19,37)(20,33)(21,35)(22,36)(23,28)(27,41)(30,42)(31,34)(32,39)(38,40);(11,14)(12,19)(13,23)(15,27)(16,30)(17,28)(18,32)(20,31)(21,38)(22,39)(24,26)(25,41)(29,42)(33,35)(34,40)(36,37);(5,10)(7,9)(11,14)(12,20)(13,24)(15,30)(16,27)(17,28)(18,33)(19,31)(21,39)(22,38)(23,26)(25,41)(29,42)(32,35)(34,37)(36,40);(4,8)(7,9)(11,15,17,16)(12,21,22,31)(13,25,26,29)(14,27,28,30)(18,34,36,35)(19,38,39,20)(23,41,24,42)(32,40,37,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484475);