Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#13350 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2249 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 2249 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,14)(6,11)(7,10)(8,13)(9,15)(12,20)(16,27)(17,29)(18,24)(19,26)(21,34)(22,30)(23,31)(25,33)(28,32);(3,5)(4,9)(6,15)(7,16)(8,17)(10,23)(11,14)(12,21)(13,22)(18,31)(19,30)(20,32)(24,27)(25,28)(26,29)(33,34);(4,10)(7,20)(9,21)(11,24)(12,13)(14,27)(15,30)(16,32)(18,33)(22,23)(25,26)(28,29)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#499825, C2×#498898, C2×#499819, … |
| maximal normal subgroup of | C2×#499825, C2×#498898, C2×#499819, … |
| maximal quotient of | #72280866, #72281488, #72281925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406090);