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 |
| prime radical | #44886 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(9,10,14,16)(11,19,22,17)(12,15,23,21)(13,24,20,18);(1,2)(3,4)(5,6)(7,8)(9,11,20,23,14,22,13,12)(10,15,24,17,16,21,18,19);(6,8)(9,10)(11,21)(12,17)(13,18)(14,16)(15,22)(19,23)(20,24);(2,4)(6,8)(10,16)(11,22)(12,23)(18,24);(9,12,20,11,14,23,13,22)(10,17,24,21,16,19,18,15)Polycyclic presentation:
2^9|a^2=i,b^2=fi,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, 7426247);