Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #347058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62931 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,7)(3,8)(4,5)(9,10,14,13)(11,15,19,18)(12,16,21,20)(17,23,24,22);(1,2,6,3)(4,8,5,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24);(9,12,14,21)(10,16,13,20)(11,17,19,24)(15,22,18,23);(1,3,6,2)(4,8,5,7)(9,13,14,10)(11,18,19,15)(12,20,21,16)(17,23,24,22)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=hi,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,e^b=eh,f^b=fi,e^c=ei,g^d=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254321);