Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153126 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13343 |
| commutator, Frattini, cosocle | C2×C4⋊Q8 | C23 |
| 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 | |
569 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 569 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)(3,15)(4,10)(5,26)(6,11)(7,13)(8,14)(9,17)(12,27)(16,49)(18,41)(19,34)(20,44)(21,57)(22,45)(23,60)(24,39)(25,40)(28,42)(29,43)(30,35)(31,47)(32,46)(33,54)(36,53)(37,52)(38,59)(48,51)(50,61)(55,62)(56,63)(58,64);(1,3)(2,9)(4,21)(5,19)(6,17)(7,30)(8,18)(10,37)(11,41)(12,44)(13,36)(14,34)(15,26)(16,23)(20,32)(22,56)(24,58)(25,54)(27,53)(28,51)(29,55)(31,50)(33,38)(35,43)(39,49)(40,63)(42,64)(45,57)(46,62)(47,48)(52,59)(60,61);(1,4)(2,10)(3,16)(5,23)(6,28)(7,31)(8,25)(9,33)(11,42)(12,45)(13,47)(14,40)(15,48)(17,21)(18,51)(19,54)(20,56)(22,27)(24,32)(26,60)(29,59)(30,52)(34,57)(35,63)(36,64)(37,44)(38,43)(39,46)(41,49)(50,53)(55,58)(61,62)Polycyclic presentation:
2^9|d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fgh,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gh,d^c=dg,f^c=fi,g^c=gi,h^c=hi,f^d=fhi,g^d=gi,g^f=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60322);