Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32631 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^9|a^2=hi,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=bh,c^a=ci,f^a=fh,g^a=gi,e^b=eh,f^b=fi,d^c=dh,e^c=ei134 conjugacy classes — view the full character table.
| Fitting subgroup of | #72262057 |
| maximal subgroup of | #72234899, #72236765, #72237255, … |
| maximal normal subgroup of | #72234899, #72236765, #72237255, … |
| maximal quotient of | #72234903, #72234928, #72238556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10494172);