Finite Group DatabasebetaFinite Group Databasebeta| order | 32576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C4⋊Q8×C509 ⋊ C2 |
| Sylow | C509 ⋊ #533 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D509 |
| commutator, socle, cosocle | C22×C509 | C24 |
| Fitting | C4⋊Q8×C509 | C2 |
| prime radical | C22×D509 | C23 |
| hypercenter | C4⋊Q8 | D509 |
| hypocenter | C509 | #533 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 16288 |
Polycyclic presentation:
2^6,509|a^2=f,c^2=f,d^2=e,b^a=be,c^a=cf,g^b=g^508,g^d=g^5083584 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32576, 1104);