Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93814 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7080 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | #2151 | C22 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2151 ◁ G | |
| LECS | G ▷ #2151 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
135 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 26 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:
(1,2,4,11,6,13,25,42,8,14,21,43,7,30,26,56)(3,9,17,31,18,33,52,60,20,34,22,39,19,46,53,64)(5,27,24,55,28,45,51,40,29,57,23,54,16,12,44,41)(10,35,36,47,37,62,58,49,38,50,63,48,32,61,59,15);(2,9)(3,15)(4,21)(5,16)(6,7)(10,12)(11,39)(13,46)(14,34)(17,50)(18,48)(19,47)(20,49)(22,35)(23,44)(24,51)(27,32)(28,29)(30,33)(31,43)(36,40)(37,57)(38,45)(41,63)(42,60)(52,62)(53,61)(54,59)(55,58)(56,64)Polycyclic presentation:
2^9|a^2=d,c^2=fh,d^2=g,e^2=hi,f^2=hi,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=defh,e^b=eh,f^b=fh,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, 1010);