Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7092 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#7096 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #331 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C82 | C23 |
| prime radical | C2×#1712 | C2 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
422 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 256 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4,5,10,6,9,13,17,7,11,14,18,15,8,12,16)(19,20)(21,24)(22,31)(23,27)(25,28)(26,29)(30,33)(32,34);(4,8)(5,12)(6,15)(9,11)(10,18)(13,14)(20,24)(21,28)(22,29)(25,26)(27,33)(30,34)(31,32)86 conjugacy classes — view the full character table.
| maximal subgroup of | #72216562, C2×#123275, #72225182, … |
| maximal normal subgroup of | #72216562, C2×#123275, #72225182, … |
| maximal quotient of | C2×#93866, #72225579, #72226852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30066);