Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#7307 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
2558 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 2558 conjugacy classes of subgroups — too many to draw. Only its 170 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)(5,9)(6,8)(7,10)(11,12)(13,16,18,15)(14,17);(3,5)(4,7,8,9)(6,10)(12,15,17,16)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#139743, C2×#144366, #72219160, … |
| maximal normal subgroup of | C2×#139743, C2×#144366, #72219160, … |
| maximal quotient of | C2×#93155, #72219248, #72219296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35203);