Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7529 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7530 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1408 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | #493 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#493 ◁ G | |
| LECS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 34 points:
(1,2);(3,4)(5,13,15,30)(6,17,20,27)(7,12)(8,11,19,29)(9,14,23,33)(10,21,26,32)(16,31,22,25)(18,28)(24,34);(3,5,7,16)(4,9,12,23)(6,18,21,34)(8,22,19,15)(10,25,27,30)(11,28,29,24)(13,17,31,26)(14,32,33,20)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#153538, C2×#153618, C2×#153614, … |
| maximal normal subgroup of | C2×#153538, C2×#153618, C2×#153614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60702);