Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12114 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12116 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
1168 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 138 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)(7,19)(8,23)(9,13)(10,16)(12,21)(15,30)(17,31)(18,29)(20,25)(22,34)(24,28)(26,33)(27,32);(3,5)(4,10)(6,15)(7,20)(8,24)(9,17)(11,25)(12,28)(13,26)(14,19)(16,21)(18,33)(22,32)(23,27)(29,30)(31,34);(3,6,9,8)(4,11,13,23)(5,15,17,24)(7,21,22,18)(10,25,26,27)(12,19,29,34)(14,30,31,28)(16,32,33,20)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#486338, C2×#485587, C2×#486082, … |
| maximal normal subgroup of | C2×#486338, C2×#485587, C2×#486082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391421);