Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#1357 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #331 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#331 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
482 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4);(5,6,7,12,8,11,15,19,9,13,16,20,17,10,14,18);(6,10)(7,14)(8,17)(11,13)(12,20)(15,16)92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216405, C4×#12310, #72216475, … |
| maximal normal subgroup of | #72216405, C4×#12310, #72216475, … |
| maximal quotient of | #72216562, AGL(1,𝔽5)×#1357 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36091);