Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #37672 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,22)(21,24);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,21)(20,24)(22,23);(1,3,4,7)(2,6,5,8)(10,14)(11,12)(15,16)(21,23);(2,5)(6,8)(18,22)(21,23)Polycyclic presentation:
2^9|c^2=ei,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,f^c=fh,g^c=gh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72308058, #72308076, #72308083, … |
| maximal normal subgroup of | #72308058, #72308076, #72308083, … |
| maximal quotient of | C2×#255094 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 162290);