Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,8)(3,7,5,4)(9,10,14,16,15,11,12,18)(13,17,23,24,22,19,20,21);(1,3,6,5)(2,7,8,4)(9,11)(10,15)(12,16)(13,21)(14,18)(17,23)(19,20)(22,24);(1,4,5,2,6,7,3,8)(9,12,15,14)(10,16,11,18)(13,20,22,23)(17,24,19,21);(1,3,6,5)(2,7,8,4)(9,13)(10,17)(11,19)(12,20)(14,23)(15,22)(16,24)(18,21)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,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=chi,d^b=de,e^b=eh,d^c=di,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132930);