Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10577301 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6648 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^9|b^2=hi,c^2=i,g^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,g^a=gi,g^b=gi,d^c=dh98 conjugacy classes — view the full character table.
| maximal subgroup of | #72240220, #72240221, #72240239, … |
| maximal normal subgroup of | #72240220, #72240221, #72240239, … |
| maximal quotient of | #72240286, #72240298, #72261307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485194);