Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,7,8)(9,10,13,15,14,17,12,18)(11,16,21,22,19,24,20,23);(1,2,4,3)(10,15)(11,19)(12,13)(16,23)(17,18)(22,24);(1,3,4,2)(9,11,12,20,14,19,13,21)(10,16,18,23,17,24,15,22);(10,17)(15,18)(16,24)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,h^b=hi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265502);