Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6443048 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1936 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,21)(13,17)(14,19)(15,18)(16,22)(23,24);(2,5)(6,7)(9,11,20,10,15,19,14,18)(12,16,23,24,22,21,13,17);(1,3,4,8)(2,6,5,7)(9,12,20,23,15,22,14,13)(10,16,11,17,18,21,19,24);(9,13,14,22,15,23,20,12)(10,17,19,16,18,24,11,21);(12,22)(13,23)(16,21)(17,24)Polycyclic presentation:
2^9|b^2=fg,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373765);