Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 1048576 |
| 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 | C2×#2404 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3992 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)(10,12)(11,15)(17,18,22,21)(19,24,23,20);(6,8)(9,10,13,12)(11,16,15,14)(17,19,22,23)(18,20,21,24);(5,6)(7,8)(10,12)(14,16)(18,21)(19,23);(1,2)(3,4)(9,11,13,15)(10,14,12,16)(17,20,22,24)(18,23,21,19);(9,12,13,10)(11,14,15,16)(17,21,22,18)(19,20,23,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72333442, #72333447, #72333473 |
| maximal normal subgroup of | #72333442, #72333447, #72333473 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4029401);