Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13561 |
| 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×D4 | C22×C8 |
| Frattini, cosocle | #408 | C23 |
| socle | C2 | C2×#1316 |
| prime radical | C2×#585 | C4 |
| LCS | G ▷ C2×D4 ▷ 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:
(1,2,5,11,7,12,25,42,8,13,26,43,6,27,30,41)(3,14,17,45,18,46,23,57,15,44,48,61,28,53,52,64)(4,19,20,54,21,10,36,37,22,38,24,58,29,39,56,63)(9,31,33,47,35,59,62,50,32,55,60,16,34,49,40,51);(1,3)(2,9)(4,16)(5,23)(6,28)(7,18)(8,15)(10,14)(11,40)(12,35)(13,32)(17,30)(19,53)(20,55)(21,51)(22,47)(24,31)(25,48)(26,52)(27,34)(29,50)(33,42)(36,49)(37,57)(38,46)(39,44)(41,60)(43,62)(45,54)(56,59)(58,61)(63,64);(3,15)(9,32)(14,44)(16,47)(17,48)(18,28)(23,52)(31,55)(33,60)(34,35)(40,62)(45,61)(46,53)(49,59)(50,51)(57,64)Polycyclic presentation:
2^9|a^2=e,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,e^b=efi,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, 56742);