Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | Q8×#379 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
18374 conjugacy classes of subgroups; only the 2042 normal subgroups are shown.
This group has 18374 conjugacy classes of subgroups — too many to draw. Only its 2042 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) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(13,14,16,17)(15,19,20,18);(13,15,16,20)(14,18,17,19);(2,4)(5,6,7,8)(10,12)(13,17,16,14)(15,19,20,18);(9,10)(11,12);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)140 conjugacy classes — view the full character table.
| maximal subgroup of | #72267497, #72267499, #72267502, … |
| maximal normal subgroup of | #72267497, #72267499, #72267502, … |
| maximal quotient of | #72267485, #72337342 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 427662);