Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #180869 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#97 |
| prime radical | C25 | M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,17,13,15,21,22,14,18,20,24,11,16,19,23);(1,3)(2,5)(4,8)(6,7)(10,15,18,16)(12,20)(17,23,24,22)(19,21);(3,8)(5,7);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,f^2=h,h^2=i,b^a=be,d^a=deh,g^a=gi,c^b=ceh,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239325, #72239337, #72239455, … |
| maximal normal subgroup of | #72239325, #72239337, #72239455, … |
| maximal quotient of | #72309885, C2×#493273, #72317183, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400469);