Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(14,17)(15,19)(18,20);(1,3,4,2)(6,9)(10,12);(13,14,15,17,16,18,19,20);(5,7,8,11)(6,10,9,12);(14,18)(17,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72285146, #72305397, #117×AGL(1,ℤ/8), … |
| maximal normal subgroup of | #72285146, #72305397, #117×AGL(1,ℤ/8), … |
| maximal quotient of | #72305574 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279319);