Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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 | C2×#3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2724 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)(3,7)(4,6)(5,8)(9,10,13,17,14,12,21,16)(11,15,20,23,19,18,24,22);(1,3)(2,5)(4,8)(6,7)(9,11,13,20,14,19,21,24)(10,15,17,23,12,18,16,22);(9,12)(10,14)(11,18)(13,16)(15,19)(17,21)(20,22)(23,24);(3,8)(5,7)(11,19)(15,18)(20,24)(22,23);(2,6)(5,7)(11,19)(15,18)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314194, #72315619, #72315633 |
| maximal normal subgroup of | #72314194, #72315619, #72315633 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6294886);