Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13569 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C8 |
| Frattini, cosocle | #391 | C23 |
| prime radical | C22×2−1+4 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1018 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19,22,37)(12,24,26,36)(13,28,30,17)(14,18)(15,21,32,39)(16,27)(20,34,23,38)(25,33,40,31)(29,35);(9,11)(10,15)(12,25)(13,22)(14,20)(16,34)(17,32)(18,31)(19,35)(21,26)(23,30)(24,38)(27,39)(28,33)(29,40)(36,37);(11,20)(15,31)(19,34)(21,25)(22,23)(32,33)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=e,d^2=i,e^2=g,f^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.
| maximal subgroup of | #72340590, #72340679, #72340681 |
| maximal normal subgroup of | #72340590, #72340679, #72340681 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56739);