Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,14)(11,16,15,12)(17,18,22,20)(19,21,24,23);(6,8)(10,14)(11,15)(17,19,22,24)(18,23,20,21);(5,6,7,8)(10,14)(12,16)(17,18,22,20)(19,21,24,23);(1,2)(3,4)(9,11,13,15)(10,12,14,16)(17,20,22,18)(19,21,24,23);(9,12)(10,15)(11,14)(13,16)(17,21)(18,19)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=h,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72333576, #72333615, #72333736 |
| maximal normal subgroup of | #72333576, #72333615, #72333736 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 898208);