Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C4 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C22 | C24×D4 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10446 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^9|a^2=h,c^2=h,g^2=i,b^a=bg,f^a=fi,g^a=gi,e^b=ei,g^b=gi,d^c=di164 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#63090, #72226742, #72281655, … |
| maximal normal subgroup of | C4×#63090, #72226742, #72281655, … |
| maximal quotient of | #72282339, #72282680, #72305273, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481362);