Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(5,6,7,8)(9,10)(11,17)(12,21)(13,16)(14,19)(15,23)(18,22)(20,24);(1,2)(3,4)(9,10)(11,18,20,14)(12,21)(13,16)(15,23)(17,22,24,19);(1,3,4,2)(9,11)(10,14)(12,22)(13,20)(15,24)(16,18)(17,21)(19,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fi,e^c=ei,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238814, #72238843 |
| maximal normal subgroup of | #72238814, #72238843 |
| maximal quotient of | #72238818 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30944);