Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C2×AGL(1,ℤ/8) | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
633 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,7,5,4,3,8)(9,10)(11,13)(12,20)(14,19)(15,17)(16,21)(18,23)(22,24);(1,3,5,6)(2,7,4,8)(9,11,13,10,15,19,14,17)(12,18,21,16,23,20,22,24);(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20);(1,4,6,8,5,2,3,7)(9,13,15,14)(10,11,17,19)(12,21,23,22)(16,18,24,20)Polycyclic presentation:
2^9|a^2=g,b^2=eh,d^2=gh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dhi,f^c=fh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 118267);