Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7422 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1347 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×#346 | C22 |
| socle | C2 | #7112 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(1,2,5,13,8,14,6,32,10,15,31,47,9,36,7,34)(3,16,18,48,20,49,35,43,21,51,19,55,37,50,33,63)(4,22,24,56,27,57,26,12,29,44,30,45,28,60,25,46)(11,23,38,53,40,59,62,54,42,52,64,58,41,61,39,17);(1,3)(2,11)(4,23)(5,19)(6,33)(7,35)(8,20)(9,37)(10,21)(12,43)(13,38)(14,40)(15,42)(16,22)(17,26)(18,31)(24,58)(25,54)(27,59)(28,61)(29,52)(30,53)(32,62)(34,39)(36,41)(44,51)(45,55)(46,63)(47,64)(48,56)(49,57)(50,60)Polycyclic presentation:
2^9|a^2=d,c^2=fhi,d^2=g,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=defhi,e^b=ehi,f^b=fhi,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225354 |
| maximal normal subgroup of | #72225354 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1122);