Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10583369 |
| 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 ◁ #3531 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5848 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=h,g^2=i,b^a=bg,c^a=ch,g^a=gi,c^b=ci,e^b=ei,g^b=gi,e^c=ei,f^c=fh,e^d=ehi,f^d=fi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72250754 |
| maximal normal subgroup of | #72250754 |
| maximal quotient of | #72257181, #72257494, #72257520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10492417);