Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #273947 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3,6,8,7,4)(9,10)(11,21)(12,16)(13,22)(14,19)(15,20)(17,23)(18,24);(1,3)(2,5)(4,6)(7,8)(9,11,15,17)(10,13,20,12)(14,16,24,22)(18,23,19,21);(1,4)(2,7)(3,6)(5,8)(9,12)(10,16)(11,18)(13,15)(14,23)(17,19)(20,22)(21,24);(1,2,6,8)(3,5,4,7)(9,13)(10,17)(11,20)(12,15)(14,23)(16,19)(18,22)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=hi,e^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fhi,g^c=gh,e^d=ei,f^d=fhi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 181162);