Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
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 | C23×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4438 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)(9,10,13,17,14,11,18,16)(12,15,21,23,20,19,24,22);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,19)(13,16)(15,20)(17,18)(21,22)(23,24);(9,12)(10,15)(11,19)(13,21)(14,20)(16,22)(17,23)(18,24);(12,20)(15,19)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=h,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,g^c=gi,e^d=ei122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278109);