Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #397 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1510 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)(3,4)(5,6,7,8)(10,15)(11,19)(12,13)(16,24)(17,18)(20,23);(9,10,13,18,14,17,12,15)(11,20,21,24,19,16,22,23);(2,4)(9,11,12,21,14,19,13,22)(10,16,15,23,17,20,18,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,f^b=fi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238743, #72238746, #72238753, … |
| maximal normal subgroup of | #72238743, #72238746, #72238753, … |
| maximal quotient of | #72239045 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30969);