Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#12066 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #442 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,6,9,7,5,8,10)(11,12,15,21,17,22,16,19)(13,23,25,40,27,41,26,37)(14,28,31,38,30,36,32,20)(18,33,34,24,35,29,39,42);(11,13,17,27)(12,18,22,35)(14,29,30,33)(15,26,16,25)(19,34,21,39)(20,37,38,40)(23,28,41,36)(24,32,42,31);(3,5,6,10,7,4,8,9)(12,19,22,21)(14,30)(18,34,35,39)(20,28,38,36)(23,37,41,40)(24,42)(29,33)(31,32)92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#483589, C2×#151105, C2×#483597, … |
| maximal normal subgroup of | C2×#483589, C2×#151105, C2×#483597, … |
| maximal quotient of | C2×#129165, C2×#140354, #72321448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390783);