Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1396 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)(9,10,14,19,15,20,13,18)(11,17,22,24,21,23,12,16);(5,6)(7,8)(9,11,13,12,15,21,14,22)(10,16,19,17,20,24,18,23);(2,4)(6,8)(9,12,13,21,15,22,14,11)(10,17,19,24,20,23,18,16);(10,18,20,19)(11,21)(12,22)(16,17,24,23)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=eh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257066, #72257156, #72257164, … |
| maximal normal subgroup of | #72257066, #72257156, #72257164, … |
| maximal quotient of | #72265472, C2×#376260, #72322667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283456);