Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#60462 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,12,16)(9,17,19,30)(10,14,22,28)(11,24,18,15)(13,25,26,21)(20,31,33,38)(23,35,32,29)(27,36,37,34);(7,9,18,25,12,19,11,21)(8,13,24,30,16,26,15,17)(10,20,32,36,22,33,23,34)(14,27,35,38,28,37,29,31);(4,6)(9,19)(13,26)(17,30)(20,33)(21,25)(27,37)(31,38)(34,36);(7,10)(8,14)(9,20)(11,23)(12,22)(13,27)(15,29)(16,28)(17,31)(18,32)(19,33)(21,34)(24,35)(25,36)(26,37)(30,38);(10,22)(14,28)(20,33)(23,32)(27,37)(29,35)(31,38)(34,36)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482111);