Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C26 |
| commutator, socle | C22 | C25×C4 |
| prime radical | C2×#585 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7225 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=g,h^2=i,b^a=bg,c^a=ci,d^a=dgi,f^a=fi,c^b=cgi,e^b=ei,e^c=ei,f^e=fi170 conjugacy classes — view the full character table.
| maximal subgroup of | M16×#585 |
| maximal normal subgroup of | M16×#585 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481853);