Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136388 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8890 |
| 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 | |
318 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 318 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)(3,8,14,32)(4,17,20,51)(5,22)(6,10)(7,28)(9,19,35,54)(11,26)(12,27)(13,41)(15,37,45,63)(16,39,29,33)(18,50,21,38)(23,31,57,62)(24,40,55,34)(25,49)(30,56)(36,53)(42,61)(43,60)(44,64)(46,58)(47,48)(52,59);(1,3,7,29,6,14,27,16)(2,8,12,39,10,32,28,33)(4,13,18,42,20,44,21,47)(5,23,26,15,25,57,53,45)(9,30,40,59,35,60,34,58)(11,37,22,31,36,63,49,62)(17,48,38,64,51,61,50,41)(19,52,24,56,54,46,55,43);(1,4)(2,9)(3,13)(5,24)(6,20)(7,18)(8,30)(10,35)(11,38)(12,40)(14,44)(15,46)(16,47)(17,49)(19,53)(21,27)(22,51)(23,56)(25,55)(26,54)(28,34)(29,42)(31,61)(32,60)(33,58)(36,50)(37,64)(39,59)(41,63)(43,57)(45,52)(48,62)Polycyclic presentation:
2^9|a^2=f,b^2=fh,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dg,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,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, 43584);