Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10,13,17,16,21,15,19)(11,12,20,14,18,23,22,24);(1,2)(3,6)(4,5)(7,8)(10,17,21,19)(11,22,18,20)(12,23)(14,24);(2,5)(6,8)(9,11,14,10,15,20,12,17,16,18,24,21,13,22,23,19);(1,3,4,7)(2,6,5,8)(9,12,13,14,16,23,15,24)(10,18,19,20,21,11,17,22)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,d^2=e,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264898, #72264975, #72265009, … |
| maximal normal subgroup of | #72264898, #72264975, #72265009, … |
| maximal quotient of | #72309731, #72309803, #72310275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404253);