Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10,13,16,14,18,12,17)(11,19,20,15,22,24,21,23);(9,11)(10,15)(12,21)(13,20)(14,22)(16,24)(17,19)(18,23);(10,16)(12,13)(15,19)(17,18)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,d^c=dh,e^c=eh,g^c=gh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281267, #72302216, #72302217 |
| maximal normal subgroup of | #72281267, #72302216, #72302217 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18759);