Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2076 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)(3,4)(10,15,18,17)(11,19,20,21);(5,6)(7,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,23)(20,24)(21,22);(2,4)(6,8)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,23)(18,24);(2,4)(10,17)(12,13)(15,18)(16,23)(19,21)(22,24)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,h^c=hi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290324);