Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #122137 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1446 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,6,4,3,7,8)(9,10)(11,14,18,22)(12,19)(13,16)(15,20)(17,21,24,23);(5,7)(6,8)(9,11)(10,14,16,22)(12,17)(13,18)(15,23,19,21)(20,24);(1,3)(2,4)(5,6)(7,8)(9,12)(10,15,16,19)(11,17)(13,20)(14,21,22,23)(18,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=eh,g^c=ghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29337);