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 |
| prime radical | #61199 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3,6,8,4,7)(9,10)(11,20)(12,15)(13,18)(14,19)(16,21)(17,24)(22,23);(1,3,6,7)(2,4,8,5)(9,11)(10,12)(13,23)(14,16)(15,24)(17,21)(18,20)(19,22);(1,2)(3,4)(5,7)(6,8)(9,12,14,22)(10,15,19,23)(11,17,16,18)(13,20,24,21);(1,2)(3,4)(5,7)(6,8)(9,12,14,22)(10,16,19,11)(13,21,24,20)(15,18,23,17)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=h,d^2=h,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, 181196);