Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1437 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,12,17,14,16,13,18)(11,19,21,24,20,15,22,23);(1,2,4,8,5,7,3,6)(9,11,13,21,14,20,12,22)(10,15,18,23,16,19,17,24);(2,7)(6,8)(9,10,12,17,14,16,13,18)(11,19,22,23,20,15,21,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,e^b=ei,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72343019, #72343034, #72343044, … |
| maximal normal subgroup of | #72343019, #72343034, #72343044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18320);