Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60688 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18,15,19,13,23)(11,20,12,17,21,24,22,16);(2,5)(6,8);(9,11,13,22,15,21,14,12)(10,16,18,20,19,17,23,24);(9,12,14,21,15,22,13,11)(10,17,18,24,19,16,23,20);(1,3,4,7)(2,6,5,8)(11,21)(12,22)(16,17)(20,24)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,e^c=ei,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239527, #72239727, #72239732, … |
| maximal normal subgroup of | #72239527, #72239727, #72239732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281173);