Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10336 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
| direct product | C2×#10337 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2403 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
4254 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 4254 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,6,8)(5,7,9,10)(11,12)(13,15)(14,16)(17,18);(7,10)(11,13)(12,15)(14,17)(16,18);(3,5)(4,7)(6,9)(8,10)(15,18)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233449, C2×#233541, #72236728, … |
| maximal normal subgroup of | #72233449, C2×#233541, #72236728, … |
| maximal quotient of | #72233441, C2×#111458, C2×#144414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140732);