Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#546 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24);(2,4)(6,8)(9,10,12,14)(11,13,16,15)(18,21)(19,23);(1,2)(3,4)(5,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19,20,23)(18,22,21,24);(9,11)(10,13)(12,16)(14,15);(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|b^2=i,c^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232223, #72232228, #72232272, … |
| maximal normal subgroup of | #72232223, #72232228, #72232272, … |
| maximal quotient of | #72250283, #72330414, #72330662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 563629);