Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93780 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7076 |
| commutator | C2×M32 | C2×C4 |
| Frattini, cosocle | #2112 | C22 |
| prime radical | #13309 | C2 |
| LCS | G ▷ C2×M32 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2112 ◁ G | |
| LECS | G ▷ #2112 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,8,14,29,13)(3,15,17,48,21,51,20,50)(4,22,25,57,27,43,62,58)(6,30,28,12,7,33,24,47)(9,35,36,53,39,52,38,16)(10,40,42,32,45,63,44,26)(18,54,31,41,19,56,34,49)(23,59,60,37,55,46,61,64);(1,3)(2,9)(4,23)(5,20)(6,31)(7,34)(8,21)(10,41)(11,36)(12,46)(13,38)(14,39)(15,43)(16,26)(17,29)(18,28)(19,24)(22,51)(25,60)(27,55)(30,37)(32,53)(33,64)(35,40)(42,54)(44,56)(45,49)(47,59)(48,57)(50,58)(52,63)(61,62)Polycyclic presentation:
2^9|a^2=d,c^2=fh,d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,h^b=hi,d^c=dghi,e^c=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 977);