Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136422 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8894 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 64 points:
(1,2,6,12,7,10,29,13)(3,8,17,32,15,33,28,34)(4,18,19,51,22,52,21,39)(5,23,26,37,27,40,54,11)(9,25,35,56,36,55,41,20)(14,42,45,50,46,64,43,63)(16,48,24,57,47,38,59,31)(30,49,61,58,62,53,60,44);(1,3)(2,8,10,33)(4,14)(5,24)(6,28)(7,15)(9,30,36,62)(11,38,37,48)(12,34,13,32)(16,26)(17,29)(18,50,52,63)(19,43)(20,53)(21,45)(22,46)(23,57,40,31)(25,58)(27,59)(35,60,41,61)(39,64,51,42)(44,55)(47,54)(49,56);(1,4)(2,9)(3,14)(5,25)(6,19)(7,22)(8,30)(10,36)(11,39)(12,35)(13,41)(15,46)(16,49)(17,45)(18,23)(20,54)(21,29)(24,58)(26,56)(27,55)(28,43)(31,63)(32,61)(33,62)(34,60)(37,51)(38,64)(40,52)(42,48)(44,59)(47,53)(50,57)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,d^b=dg,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43619);