Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #393 |
| 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:
(1,2)(3,4)(9,10,15,17)(11,18)(12,14,23,13)(16,21)(19,22)(20,24);(5,6)(7,8)(9,11,14,21,15,22,13,20)(10,16,12,18,17,24,23,19);(2,4)(6,8)(9,12,13,17,15,23,14,10)(11,19,20,16,22,18,21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,h^a=hi,c^b=cf,f^b=fi,g^b=gi,e^c=ei,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235145, #72235149, #72235153, … |
| maximal normal subgroup of | #72235145, #72235149, #72235153, … |
| maximal quotient of | C2×#123745, #72235161, #72235192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30941);