Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C227 |
| extraspecial | 2−1+8 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C28 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
419155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^9|a^2=i,b^2=i,b^a=bi,d^c=di,f^e=fi,h^g=hi257 conjugacy classes — view the full character table.
| Fitting subgroup of | #72268003, #72268574, #72268576 |
| maximal subgroup of | C2×2−1+8, #72258201, #72263037, … |
| maximal normal subgroup of | C2×2−1+8, #72258201, #72263037, … |
| maximal quotient of | C2×2−1+8 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10494212);