Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #253411 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #18316 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(11,17)(12,20)(14,23)(15,21)(18,24)(19,22);(3,8)(5,6)(9,10,13,16)(11,18)(12,21,20,15)(14,19)(17,23)(22,24);(2,5)(3,8)(6,7)(9,11)(10,14)(12,22)(13,19)(15,24)(16,18)(17,20)(21,23);(12,20)(15,21)(17,22)(23,24)Polycyclic presentation:
2^9|b^2=eh,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^c=fi,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72284437 |
| maximal normal subgroup of | #72284437 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 160613);