Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #127 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1445 |
| prime radical | C24 | C4×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10,15,12,14,18,13,17)(11,19,22,16,23,24,21,20);(9,11,15,21,14,23,13,22)(10,16,12,19,18,20,17,24);(9,12,15,18,14,17,13,10)(11,20,21,24,23,16,22,19)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,f^c=fh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72226705, #72226711, #72226712, … |
| maximal normal subgroup of | #72226705, #72226711, #72226712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28820);