Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246079 |
| 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 | #9914 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1349 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(18,21)(19,23)(22,24);(13,15)(17,18)(19,24)(20,22)(21,23);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(18,22)(21,24)Polycyclic presentation:
2^9|d^2=eh,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,g^c=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72308003, #72308044, #72308058, … |
| maximal normal subgroup of | #72308003, #72308044, #72308058, … |
| maximal quotient of | C2×#258135 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 165331);