Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60595 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7510 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)(5,8)(6,7)(9,10)(11,12)(13,22)(14,25)(15,18)(16,20)(17,31)(19,28)(21,30)(23,39)(24,34)(26,42)(27,36)(29,38)(32,35)(33,41)(37,40);(4,7)(8,10)(11,13,16,15)(12,18,20,22)(14,23,27,26)(17,24,30,29)(19,33,35,37)(21,34,31,38)(25,42,36,39)(28,40,32,41);(11,14,16,27)(12,19,20,35)(13,23,15,26)(17,32,30,28)(18,33,22,37)(21,36,31,25)(24,41,29,40)(34,39,38,42);(3,5,6,9)(4,8,7,10)(11,15,16,13)(12,18,20,22)(14,26,27,23)(17,29,30,24)(19,33,35,37)(21,34,31,38)(25,42,36,39)(28,41,32,40);(13,15)(18,22)(23,26)(24,29)(33,37)(34,38)(39,42)(40,41)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482326);