Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153145 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13346 |
| commutator, Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #33569 | C2 |
| LCS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
539 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(1,2,7,12)(3,8,14,31)(4,9,21,38)(5,22,26,41)(6,27,29,11)(10,25,42,28)(13,43,45,30)(15,48,23,59)(16,51,47,33)(17,53,49,64)(18,37,20,35)(19,57,24,60)(32,50,39,52)(34,55,63,54)(36,58,40,56)(44,61,46,62);(1,3)(2,8)(4,17)(5,23)(6,16)(7,14)(9,34)(10,39)(11,33)(12,31)(13,19)(15,26)(18,55)(20,54)(21,49)(22,59)(24,45)(25,52)(27,51)(28,50)(29,47)(30,36)(32,42)(35,64)(37,53)(38,63)(40,43)(41,48)(44,56)(46,58)(57,61)(60,62);(1,4)(2,9)(3,13)(5,24)(6,18)(7,21)(8,30)(10,40)(11,35)(12,38)(14,45)(15,49)(16,44)(17,23)(19,26)(20,29)(22,60)(25,56)(27,37)(28,58)(31,43)(32,63)(33,61)(34,39)(36,42)(41,57)(46,47)(48,53)(50,55)(51,62)(52,54)(59,64)Polycyclic presentation:
2^9|a^2=i,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fghi,g^a=gi,c^b=cf,d^b=dh,e^b=ei,f^b=fi,g^b=gh,d^c=dg,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60342);