Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | #466 | C8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
243 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 256 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,12,15,29)(4,8)(5,19)(6,11)(7,14,23,31)(9,17)(10,18)(13,28,16,25)(20,27,32,22)(21,24)(26,30)(33,34,36,41,38,42,49,56,37,40,48,53,50,54,61,63)(35,43,45,55,46,57,59,64,44,39,47,51,58,52,60,62);(1,3)(2,7)(4,13)(5,20)(6,15)(8,22)(9,25)(10,27)(11,23)(12,24)(14,30)(16,18)(17,31)(19,29)(21,32)(26,28)(33,35)(34,39)(36,47)(37,44)(38,46)(40,43)(41,55)(42,52)(45,48)(49,60)(50,58)(51,53)(54,57)(56,64)(59,61)(62,63);(1,4,6,18)(2,8,11,10)(3,13,15,16)(5,17,21,30)(7,22,23,27)(9,24,26,19)(12,28,29,25)(14,20,31,32)(35,44)(39,43)(45,47)(46,58)(51,55)(52,57)(59,60)(62,64)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=fi,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ci,d^b=dfhi,e^b=ehi,f^b=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29714);