Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #10585860 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C24×D8 |
| commutator, Frattini, cosocle | C8 | C26 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #585 ◁ #3573 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
7898 conjugacy classes of subgroups; only the 3440 normal subgroups are shown.
This group has 7898 conjugacy classes of subgroups — too many to draw. Only its 3440 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^9|g^2=hi,h^2=i,b^a=bg,g^a=gh,h^a=hi,g^b=gh,h^b=hi,f^c=fi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72319485, #72343953, #72343960 |
| maximal normal subgroup of | #72319485, #72343953, #72343960 |
| maximal quotient of | #72343825, #72343834, #72343839 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493056);