Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93802 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7079 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | #2113 | C22 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ #2113 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
117 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 117 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:
(1,2,4,12,9,16,29,47)(3,17,19,43,23,51,25,57)(5,14,31,45,32,50,26,59)(6,33,27,44,30,13,48,49)(7,15,24,46,8,34,28,60)(10,35,36,21,39,56,62,22)(11,40,41,53,37,55,63,20)(18,42,52,64,54,38,58,61);(2,10,16,39)(3,18)(4,24)(5,30)(6,32)(7,8)(11,14,37,50)(12,43,47,57)(13,38,33,42)(15,51,34,17)(19,55)(20,21)(22,53)(23,54)(25,40)(27,48)(28,29)(35,58)(36,60,62,46)(41,44,63,49)(45,61,59,64)(52,56)Polycyclic presentation:
2^9|a^2=d,b^2=gh,c^2=fh,d^2=i,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=defhi,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,d^c=dghi,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 999);