Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12214 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1691 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1691 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) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20);(4,7)(8,10)(11,13,15,21)(12,16,18,25)(14,23,24,20)(17,26,22,19);(5,9)62 conjugacy classes — view the full character table.
| maximal subgroup of | #72209872 |
| maximal normal subgroup of | #72209872 |
| maximal quotient of | #72209872, #72227838, #72227843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391507);