Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213564 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(9,10)(11,15)(12,21,19,16)(13,22,23,17)(14,18)(20,24);(1,2)(3,4)(5,6)(7,8)(9,11,14,20)(10,15,18,24)(12,13,19,23)(16,17,21,22);(2,4)(9,12,14,19)(10,16,18,21)(11,13,20,23)(15,17,24,22);(6,8)(9,13)(10,17)(11,19)(12,20)(14,23)(15,21)(16,24)(18,22)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,h^c=hi,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72313951, #72313958, #72313993 |
| maximal normal subgroup of | #72313951, #72313958, #72313993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290268);