Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238849 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60916 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
977 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)(3,6)(4,7)(5,8)(9,10)(11,18)(12,14)(13,16)(15,19)(17,21)(20,23)(22,24);(1,2)(3,7)(4,6)(5,8)(10,14)(11,19)(12,21)(15,23)(16,17)(18,22);(2,6)(3,4)(7,8)(9,11,12,20,13,19,21,24)(10,15,17,23,16,18,14,22);(10,16)(11,19)(14,17)(20,24)Polycyclic presentation:
2^9|c^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=gi,d^c=dh,f^c=fhi,g^c=ghi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72305442, #72305449, #72305489 |
| maximal normal subgroup of | #72305442, #72305449, #72305489 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146045);