Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45355 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,7)(10,16)(11,19)(12,14)(13,18)(17,24)(20,23);(2,4)(6,7)(9,10,14,16,15,13,12,18)(11,20,21,24,19,17,22,23);(5,6,8,7)(9,11,15,19)(10,17,13,20)(12,22,14,21)(16,23,18,24);(1,2,3,4)(5,7,8,6)(9,12,15,14)(10,18,13,16)(11,21,19,22)(17,23,20,24);(9,13,14,18,15,10,12,16)(11,20,21,24,19,17,22,23)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=hi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261100, #72261157, #72315902, … |
| maximal normal subgroup of | #72261100, #72261157, #72315902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521224);