Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152981 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13239 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#2263 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
270 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,7,13)(3,16,11,17)(4,22)(5,29,31,20)(6,33,32,54)(8,39,37,19)(9,40,38,15)(10,41)(12,47,21,36)(14,35,34,30)(18,50)(23,28)(24,45)(25,46)(26,53)(27,43)(42,56)(44,49)(48,55)(51,52)(57,64)(58,60)(59,63)(61,62);(1,3,17,2,8,20,32,14,38,12,47,40,35,33,5,19,7,11,16,13,37,29,6,34,9,21,36,15,30,54,31,39)(4,23,48,63,26,58,25,52,50,18,51,46,60,53,59,55,28,22,49,45,61,56,41,57,27,43,64,10,42,62,24,44);(1,4,16,48,8,26,6,25,38,50,36,51,35,60,31,59,7,28,17,49,37,61,32,41,9,27,47,64,30,42,5,24)(2,10,20,53,14,44,12,22,40,63,33,56,19,52,11,43,13,46,29,62,34,55,21,23,15,45,54,58,39,57,3,18)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=egh,d^2=gh,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225562, #72225566, #72225574, … |
| maximal normal subgroup of | #72225562, #72225566, #72225574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60177);