Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C2×#585 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)Polycyclic presentation:
2^9|a^2=g,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^b=fh,g^b=gh,e^c=eh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72315960, #72315972, #72315990, … |
| maximal normal subgroup of | #72315960, #72315972, #72315990, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92102);