Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5498 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)(6,7)(9,10,13,14)(11,12,16,15)(17,18,20,21)(19,22,24,23);(10,14)(12,15)(17,18)(19,23)(20,21)(22,24);(5,6,8,7)(10,14)(11,16)(18,21)(22,23);(1,2,3,4)(5,7,8,6)(9,11,13,16)(10,15,14,12)(17,18)(19,22)(20,21)(23,24);(9,12,13,15)(10,16,14,11)(17,19,20,24)(18,22,21,23)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72261007, #72261048, #72261052, … |
| maximal normal subgroup of | #72261007, #72261048, #72261052, … |
| maximal quotient of | #72298132, #72298208, #72298211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 627437);