Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #63091 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C24×D4 |
| commutator | C4 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C2 | C24×C22⋊C4 |
| prime radical | #63103 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
22279 conjugacy classes of subgroups; only the 7728 normal subgroups are shown.
This group has 22279 conjugacy classes of subgroups — too many to draw. Only its 7728 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=bg,g^a=gi,g^b=gi,f^c=fi,e^d=ei164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481261);