Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C28 |
| direct product | C22×#1270 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
3262 conjugacy classes of subgroups; only the 698 normal subgroups are shown.
This group has 3262 conjugacy classes of subgroups — too many to draw. Only its 698 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2);(3,4);(5,6)(7,11,12,9)(8,10)(13,14)(15,17)(16,19)(18,20)(21,22,23,25,24,26,27,28);(5,7)(6,9)(8,12)(10,11)(14,17)(15,20)(18,19)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#102905, #72215007, #72215344, … |
| maximal normal subgroup of | C2×#102905, #72215007, #72215344, … |
| maximal quotient of | C22×#7114, C22×#7115, C2×#124066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 67613);