Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#60433 |
| 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 | |
10398 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);(4,6)(7,8)(9,17)(10,15)(11,13)(12,24)(14,20)(16,23)(18,31)(19,30)(21,27)(22,35)(25,26)(28,33)(29,34)(32,38)(36,37);(3,4,5,6)(8,13)(14,26)(15,27)(16,24)(17,30)(28,37)(29,35)(31,38);(7,9,11,19)(8,14,13,26)(10,18,21,32)(12,25,23,20)(15,28,27,37)(16,30,24,17)(22,36,34,33)(29,38,35,31);(7,10)(8,15)(9,18)(11,21)(12,22)(13,27)(14,28)(16,29)(17,31)(19,32)(20,33)(23,34)(24,35)(25,36)(26,37)(30,38);(10,21)(15,27)(18,32)(22,34)(28,37)(29,35)(31,38)(33,36)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482151);