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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2817 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)(9,10)(11,19)(12,21)(13,14)(15,17)(16,20)(18,23)(22,24);(2,4)(6,7)(9,11,14,20,15,21,23,22)(10,16,18,19,17,24,13,12);(5,6,8,7)(9,12,15,19)(10,11,17,21)(13,22,18,20)(14,16,23,24);(1,2,3,4)(5,7,8,6)(10,17)(12,19)(13,18)(16,24);(9,13,23,17,15,18,14,10)(11,12,22,24,21,19,20,16)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=h,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,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, 7521275);