Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5712 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,d^c=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72300463 |
| maximal normal subgroup of | #72300463 |
| maximal quotient of | #72300513 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10459998);