Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10)(11,20)(12,22)(13,18)(14,21)(15,19)(16,17)(23,24);(5,6)(7,8)(9,11,13,22,15,23,14,17)(10,16,21,24,19,12,18,20);(2,4)(6,8)(9,12)(10,17)(11,21)(13,20)(14,24)(15,16)(18,23)(19,22);(5,6)(7,8)(9,13,15,14)(10,18,19,21)(11,22,23,17)(12,24,16,20);(6,8)(9,13,15,14)(10,18,19,21)(11,17,23,22)(12,20,16,24)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481525);