Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #502081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,13,11,14,10,9)(7,12,16,15,18,17,20,19);(2,4)(5,7,8,16,11,18,10,20)(6,12,13,15,14,17,9,19);(6,9)(7,15)(8,10)(12,16)(13,14)(17,20)(18,19);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=f,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250979, #72252775, #72252782, … |
| maximal normal subgroup of | #72250979, #72252775, #72252782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409277);