Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7313 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#7314 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1304 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1996 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1996 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4)(5,13)(6,11,7,10)(8,12)(9,14,17,16)(15,18);(3,5,7,14,8,15,6,16)(4,9,11,18,12,17,10,13)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#139731, C2×#144382, #72218313, … |
| maximal normal subgroup of | C2×#139731, C2×#144382, #72218313, … |
| maximal quotient of | #72218313, #72236526, #72236875, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35215);