Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5666 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)(3,4)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,23)(22,24);(10,13)(14,16);(9,11,12,15)(10,14,13,16)(18,21)(19,24)(22,23);(2,4)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237127, #72237130, #72237157, … |
| maximal normal subgroup of | #72237127, #72237130, #72237157, … |
| maximal quotient of | #72249950, #72257030, #72264084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299412);