Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152540 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13012 |
| 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 | |
314 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 314 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:
(1,2,6,13,39,22,35,57,8,14,5,31,34,23,53,58,9,15,36,32,10,41,44,19,17,21,37,12,7,3,16,20)(4,24,27,47,40,51,62,11,29,46,26,18,43,50,63,48,30,45,59,55,61,54,33,49,42,64,25,56,28,60,38,52);(1,3,17,22,9,23,8,41)(2,10,14,34,15,39,21,7)(4,18,29,47,30,56,42,55)(5,32,6,12,37,13,36,31)(11,43,52,61,49,28,48,40)(16,20,44,57,53,58,35,19)(24,59,64,25,45,27,46,26)(33,54,38,50,62,51,63,60);(1,4)(2,11)(3,18)(5,33)(6,38)(7,40)(8,42)(9,30)(10,43)(12,50)(13,51)(14,52)(15,49)(16,27)(17,29)(19,45)(20,46)(21,48)(22,47)(23,56)(24,57)(25,35)(26,44)(28,39)(31,60)(32,54)(34,61)(36,63)(37,62)(41,55)(53,59)(58,64)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=ghi,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eg,f^b=fg,g^b=gh,h^b=hi,d^c=dg,e^c=egi,f^c=fg,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225559, #72225566, #72225572, … |
| maximal normal subgroup of | #72225559, #72225566, #72225572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59737);