Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #476 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8845 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
294 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2,6,14,8,15,30,13,9,16,29,12,7,33,5,28)(3,17,19,40,21,47,20,38,22,45,31,10,34,37,57,39)(4,23,24,43,26,41,25,44,27,48,32,42,35,46,59,11)(18,50,52,60,54,62,64,58,55,63,56,49,53,36,51,61);(1,3,5,20,7,34,29,19,9,22,30,57,8,21,6,31)(2,10,13,45,33,38,14,47,16,40,28,17,15,39,12,37)(4,18,25,51,35,53,24,56,27,55,59,64,26,54,32,52)(11,36,46,60,42,63,48,61,44,62,41,49,43,50,23,58);(1,4)(2,11)(3,18)(5,25)(6,32)(7,35)(8,26)(9,27)(10,36)(12,23)(13,46)(14,48)(15,43)(16,44)(17,49)(19,56)(20,51)(21,54)(22,55)(24,29)(28,41)(30,59)(31,52)(33,42)(34,53)(37,58)(38,63)(39,50)(40,62)(45,60)(47,61)(57,64)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dg,e^b=eg,f^b=fi,g^b=gi,d^c=dg,e^c=eg,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43107);