Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13569 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C8 |
| Frattini, cosocle | #408 | C23 |
| socle | C2 | C2×#1316 |
| prime radical | C2×#585 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
995 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 995 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:
(1,2,5,11,7,12,26,44,8,13,27,45,6,28,31,43)(3,14,17,46,18,47,52,63,15,38,48,61,29,59,24,56)(4,19,21,53,22,10,37,39,23,40,25,57,30,41,55,64)(9,32,34,16,36,49,42,51,33,58,60,20,35,54,62,50);(1,3)(2,9)(4,20)(5,24)(6,29)(7,18)(8,15)(10,38)(11,42)(12,36)(13,33)(14,41)(16,23)(17,26)(19,47)(21,32)(22,50)(25,58)(27,52)(28,35)(30,51)(31,48)(34,43)(37,49)(39,63)(40,59)(44,60)(45,62)(46,53)(54,55)(56,64)(57,61);(3,15)(9,33)(14,38)(16,20)(17,48)(18,29)(24,52)(32,58)(34,60)(35,36)(42,62)(46,61)(47,59)(49,54)(50,51)(56,63)Polycyclic presentation:
2^9|a^2=e,d^2=i,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=di,e^b=ef,f^b=fi,h^b=hi,e^d=ehi,f^d=fi,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56744);