Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10574841 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×#120 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
21765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^9|g^2=i,h^2=i,b^a=bg,c^a=ch,d^a=di,e^a=ei,g^a=gi,h^a=hi,f^b=fi,g^b=gi,h^c=hi,e^d=ei113 conjugacy classes — view the full character table.
| maximal subgroup of | #72302919, #72302993, #72303048, … |
| maximal normal subgroup of | #72302919, #72302993, #72303048, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482045);