Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1437 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8)(3,6,7,4)(9,10)(11,18,20,23)(12,21)(13,16)(14,17)(15,19,24,22);(1,3,6,2,5,7,4,8)(9,11)(10,15)(12,18)(13,22)(14,20)(16,23)(17,24)(19,21);(9,12,21,10,14,16,13,17)(11,18,19,15,20,23,22,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efi,d^2=i,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224699, #72224701, #72224702, … |
| maximal normal subgroup of | #72224699, #72224701, #72224702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18211);