Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152301 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12835 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
230 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 96 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)(3,16,21,48)(4,24,27,43)(5,28)(6,13)(7,35)(8,38)(9,15)(10,19,41,55)(11,26,42,34)(12,31)(14,36)(17,50)(18,52)(20,47,23,59)(22,25,57,62)(29,46)(30,39)(32,64)(33,63)(37,44)(40,54,60,56)(45,51)(49,53,61,58);(1,3,17,11,7,22,5,19,6,20,31,24,29,53,30,54,9,21,52,42,37,57,33,55,8,23,51,43,36,58,32,56)(2,10,39,25,14,34,12,16,38,60,28,49,44,27,50,59,15,41,64,62,46,26,45,48,13,40,63,61,35,4,18,47);(1,4)(2,11)(3,18)(5,25)(6,34)(7,10)(8,26)(9,27)(12,20)(13,43)(14,19)(15,42)(16,31)(17,47)(21,50)(22,39)(23,45)(24,38)(28,53)(29,60)(30,49)(32,61)(33,62)(35,56)(36,40)(37,41)(44,54)(46,55)(48,51)(52,59)(57,64)(58,63)Polycyclic presentation:
2^9|a^2=f,b^2=dgh,d^2=ghi,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,f^b=fi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225564, #72225566, #72225574, … |
| maximal normal subgroup of | #72225564, #72225566, #72225574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59497);