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×#532 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10,14,15)(11,16,12,13)(17,18)(19,24)(20,21)(22,23);(10,15)(13,16)(18,23)(19,21);(5,6)(7,8)(9,11,14,12)(10,16,15,13)(17,19,22,21)(18,24,23,20);(1,2)(3,4)(5,6)(7,8)(9,12,14,11)(10,13,15,16)(17,20)(18,19)(21,23)(22,24);(9,13)(10,11)(12,15)(14,16)(17,21,22,19)(18,20,23,24)Polycyclic presentation:
2^9|a^2=fh,c^2=f,d^2=fh,e^2=h,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 | #72311684, #72311774 |
| maximal normal subgroup of | #72311684, #72311774 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 562751);