Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(9,10,15,18)(11,19,21,12)(13,16,23,24)(14,22,20,17);(1,2)(3,4)(5,6)(7,8)(9,11,20,23,15,21,14,13)(10,16,22,12,18,24,17,19);(6,8)(9,12,15,19)(10,11,18,21)(13,22,23,17)(14,16,20,24);(2,4)(6,8)(11,21)(12,19)(13,23)(16,24);(9,13,20,11,15,23,14,21)(10,12,22,24,18,19,17,16)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426251);