Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458125 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10,13,16,12,14,22,17)(11,18,19,23,20,15,21,24);(1,2,4,5)(3,6,7,8)(10,14)(11,19,20,21)(15,23,18,24)(16,17);(1,3,4,7)(2,6,5,8)(9,11,12,20)(10,15,14,18)(13,21,22,19)(16,23,17,24);(3,7)(6,8)(10,16,14,17)(11,20)(15,24,18,23)(19,21)Polycyclic presentation:
2^9|a^2=g,b^2=efgh,c^2=e,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
| maximal quotient of | #72235489, #72235491, #72235493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381457);