Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C43 · Schur multiplier (H2) = C26×C43 |
| direct product | C2×#13750 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator | C2 | C22×C43 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6674 conjugacy classes of subgroups; only the 4354 normal subgroups are shown.
This group has 6674 conjugacy classes of subgroups — too many to draw. Only its 4354 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(7,8,10,9)(15,16,19,17)(18,21,22,20);(11,12,14,13)(16,17)(20,21);(3,4,5,6)(7,9,10,8)(11,13,14,12)(15,17,19,16)(18,21,22,20);(15,18,19,22)(16,20,17,21)272 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | C2×#13750⋊C3, #72268676 |
| maximal subgroup of | C2×#110946, C2×#110638, C2×#110637, … |
| maximal normal subgroup of | C2×#110946, C2×#110638, C2×#110637, … |
| maximal quotient of | C2×#153709 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420524);