Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232951 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1334 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,17)(13,22)(14,18)(16,19)(20,24)(21,23);(2,5)(3,8)(6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23);(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)Polycyclic presentation:
2^9|d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,g^b=gh,f^c=fi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140156);