Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#60831 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7502 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);(3,4)(5,9)(6,8)(7,10)(11,12)(13,22)(14,26)(15,21)(16,18)(17,30)(19,24)(20,28)(23,38)(25,34)(27,36)(29,35)(31,32)(33,39)(37,40)(41,42);(3,5,6,10)(4,7,8,9)(12,18)(19,32)(20,26)(21,30)(22,34)(33,42)(35,36)(37,38);(5,10)(7,9)(11,13,16,25)(12,19,18,32)(14,23,28,40)(15,24,17,31)(20,33,26,42)(21,34,30,22)(27,39,29,41)(35,37,36,38);(4,8)(7,9)(11,14)(12,20)(13,23)(15,29)(16,28)(17,27)(18,26)(19,33)(21,36)(22,37)(24,41)(25,40)(30,35)(31,39)(32,42)(34,38);(11,15,16,17)(12,21,18,30)(13,24,25,31)(14,27,28,29)(19,34,32,22)(20,35,26,36)(23,39,40,41)(33,37,42,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482734);