Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
56596 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^9|a^2=i,b^2=h,g^2=i,b^a=bh,c^a=ci,g^a=gi,f^c=fi,e^d=ei164 conjugacy classes — view the full character table.
| Fitting subgroup of | #72342107, #72342146 |
| maximal subgroup of | #72282029, #72303035, #72303070, … |
| maximal normal subgroup of | #72282029, #72303035, #72303070, … |
| maximal quotient of | #72306454 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493299);