Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1677 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9221 |
| prime radical | #33594 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 356 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:
(1,2,6,14,8,15,31,13,9,16,30,45,7,12,5,28)(3,10,20,36,21,38,32,35,22,39,29,59,18,37,19,53)(4,23,26,40,24,44,55,42,27,46,58,11,33,41,25,43)(17,47,50,61,52,62,57,64,48,63,56,34,51,54,49,60);(1,3,6,32,8,18,31,20,9,22,30,19,7,21,5,29)(2,10,13,36,12,38,14,35,16,39,28,59,15,37,45,53)(4,17,25,56,33,52,58,49,27,48,55,50,24,51,26,57)(11,34,44,63,43,64,46,62,40,61,41,47,42,60,23,54);(1,4)(2,11)(3,17)(5,26)(6,25)(7,24)(8,33)(9,27)(10,34)(12,43)(13,44)(14,46)(15,42)(16,40)(18,52)(19,50)(20,49)(21,51)(22,48)(23,45)(28,41)(29,57)(30,55)(31,58)(32,56)(35,62)(36,63)(37,60)(38,64)(39,61)(47,59)(53,54)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=gh,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=eh,f^c=fhi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45862);