Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3778 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)(11,15)(12,14)(13,16)(18,22)(19,20)(23,24);(9,11)(10,13)(12,16)(14,15);(2,4)(11,16)(13,15)(17,18,20,23,21,24,19,22);(10,14)(13,15)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,h^a=hi,c^b=ce128 conjugacy classes — view the full character table.
| maximal subgroup of | #72310237, #72310240, #72310241, … |
| maximal normal subgroup of | #72310237, #72310240, #72310241, … |
| maximal quotient of | #72317252, #72317259 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263099);