Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8610 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #380 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C4⋊#95 |
| prime radical | C24 | #117 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,15)(12,21)(13,18)(14,19)(16,20)(17,23)(22,24);(2,5)(3,7)(9,11,13,12,14,20,23,22)(10,15,18,21,19,16,17,24);(9,12,13,20,14,22,23,11)(10,16,17,21,19,15,18,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=ci,f^b=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264879, #72264886, #72264890, … |
| maximal normal subgroup of | #72264879, #72264886, #72264890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16252);