Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #22135 |
| prime radical | #3050 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,4,20,8,12,21,26,5,22,14,29)(3,13,18,28,19,24,15,30,17,27,31,32,16,9,23,25);(1,3,14,23,5,16,21,31,8,17,4,15,7,19,6,18)(2,9,20,28,22,27,10,25,12,24,29,32,11,13,26,30);(9,24)(13,27)(25,30)(28,32);(1,4,5,6,8,14,7,21)(2,10,11,20,12,26,22,29)(3,15,16,18,17,23,19,31)(9,25,13,28,24,30,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ceh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224946, #72224947, #72224959, … |
| maximal normal subgroup of | #72224946, #72224947, #72224959, … |
| maximal quotient of | C2×#496949 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404145);