Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(9,10,12,17,13,16,20,24)(11,15,19,23,18,14,21,22);(5,6)(7,8)(9,11)(10,14,16,15)(12,19)(13,18)(17,22,24,23)(20,21);(1,2,3,4)(6,8)(9,11)(10,15)(12,19)(13,18)(14,16)(17,23)(20,21)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,g^2=i,b^a=bd,c^a=ce,f^a=fi,h^a=hi,c^b=cf,d^b=di,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72215689, #72215966, #72238740, … |
| maximal normal subgroup of | #72215689, #72215966, #72238740, … |
| maximal quotient of | #72215655, #72239018, #72240268, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31216);