Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186155 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62833 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10,14,18,15,19,13,17)(11,20,21,12,16,23,22,24);(1,3)(2,6)(4,8)(5,7)(9,11,15,16)(10,12,19,24)(13,21,14,22)(17,23,18,20);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,20)(14,23)(15,24)(17,21)(18,22);(1,4,6,5)(2,7,3,8)(9,13,15,14)(10,17,19,18)(11,21,16,22)(12,23,24,20)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=i,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=ghi,d^c=di,f^c=fh,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93527);