Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2128 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| socle | C2 | #11873 |
| prime radical | #24287 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,6,12)(3,15,18,48)(4,21,22,54)(5,10,26,39)(7,14,30,44)(8,31,29,59)(9,33,35,51)(11,40,41,27)(13,46,45,25)(16,36,50,62)(17,47,52,43)(19,49,53,63)(20,38,28,57)(23,42,56,64)(24,37,55,61)(32,34,58,60);(1,3)(2,9)(4,16)(5,24)(6,28)(7,19)(8,32)(10,38)(11,34)(12,43)(13,47)(14,37)(15,42)(17,23)(18,29)(20,25)(21,49)(22,55)(26,52)(27,51)(30,58)(31,36)(33,56)(35,59)(39,48)(40,50)(41,63)(44,62)(45,61)(46,53)(54,57)(60,64);(1,4,8,5,7,23,25,27)(2,10,13,21,14,11,31,42)(3,16,20,51,19,17,32,24)(6,22,29,26,30,56,46,40)(9,34,36,49,37,38,47,15)(12,39,45,54,44,41,59,64)(18,50,28,33,53,52,58,55)(35,60,62,63,61,57,43,48)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=gi,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=dh,f^c=fi,e^d=eh,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57996);