Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93846 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7088 |
| commutator | C2×M32 | C2×C4 |
| Frattini, cosocle | #2150 | C22 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C2×M32 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2150 ◁ G | |
| LECS | G ▷ #2150 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
136 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 136 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)(3,17,19,49,22,52,36,51)(4,23,26,47)(5,31)(6,12)(7,15)(8,14)(9,16)(10,18,37,53,41,55,40,64)(11,27,43,33)(13,25)(20,56,32,50,35,42,21,58)(24,60,57,63,61,39,54,38)(28,46,62,45)(29,59,30,44)(34,48);(1,3)(2,10)(4,24)(5,32)(6,35)(7,19)(8,36)(9,22)(11,42)(12,39)(13,38)(14,40)(15,37)(16,41)(17,44)(18,28)(20,34)(21,25)(23,49)(26,61)(27,53)(29,57)(30,54)(31,63)(33,64)(43,56)(45,50)(46,58)(47,51)(48,60)(52,59)(55,62)Polycyclic presentation:
2^9|a^2=d,c^2=fhi,d^2=gh,e^2=hi,f^2=h,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=fhi,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, 1042);