Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#106 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,8,6,7,5,3)(9,10,13,17,15,16,23,19)(11,14,20,24,18,12,21,22);(1,3,5,7,6,8,4,2)(9,11,14,17,13,21,24,10,15,18,12,19,23,20,22,16);(2,7)(3,8)(10,16)(11,18)(17,19)(20,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,c^b=ch,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#126882, #72214246, #72216543, … |
| maximal normal subgroup of | C2×#126882, #72214246, #72216543, … |
| maximal quotient of | C2×#126882 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34078);