Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C25 | C24 |
| commutator | C23 | C22×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,8)(13,14,16,15)(17,18)(19,22)(20,21)(23,24);(2,4)(6,8)(9,10,11,12)(13,15,16,14)(18,21)(22,23);(1,2)(3,4)(5,6,7,8)(9,10)(11,12)(17,18)(19,23)(20,21)(22,24);(17,19,20,24)(18,22,21,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de128 conjugacy classes — view the full character table.
| maximal subgroup of | #72298108, #72298203, #72298209, … |
| maximal normal subgroup of | #72298108, #72298203, #72298209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 61105);