Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #447 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8947 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 353 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)(3,15,20,36,22,48,21,44)(4,23,26,40,28,58,27,57)(5,11)(6,32)(7,13)(8,14)(9,18,37,54,39,33,38,19)(10,25,41,34,43,29,42,24)(12,16)(17,49)(30,45)(31,46)(35,52)(47,59)(50,61)(51,56)(53,62)(55,63)(60,64);(1,3,16,18,7,20,31,54,8,22,6,33,30,21,5,19)(2,9,12,15,13,37,46,36,14,39,32,48,45,38,11,44)(4,17,24,51,26,52,25,59,28,53,34,60,27,55,29,50)(10,35,40,47,41,62,58,64,43,63,57,61,42,49,23,56);(1,4)(2,10)(3,17)(5,29)(6,34)(7,26)(8,28)(9,35)(11,23)(12,40)(13,41)(14,43)(15,47)(16,24)(18,51)(19,50)(20,52)(21,55)(22,53)(25,31)(27,30)(32,57)(33,60)(36,64)(37,62)(38,49)(39,63)(42,45)(44,56)(46,58)(48,61)(54,59)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=egi,f^c=fg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43874);