Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(10,14)(11,17)(12,20)(16,22)(18,19)(21,24);(2,4)(9,10,12,14,13,16,20,22)(11,15,18,21,19,23,17,24);(9,11)(10,15)(12,18)(13,19)(14,21)(16,23)(17,20)(22,24);(5,7,8,6)(10,16)(14,22)(15,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72323036 |
| maximal normal subgroup of | #72323036 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261436);