Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #384 | C23 |
| prime radical | #3493 | C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G | |
| LECS | G ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(2,5,6,7)(9,10)(11,20,22,30,23,35,27,15)(12,16)(13,17)(14,19)(18,28)(21,31,24,37,36,40,38,29)(25,32)(26,34)(33,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,24)(13,27)(14,23)(16,31)(17,20)(18,35)(19,30)(21,25)(22,28)(26,38)(29,33)(32,37)(34,40)(36,39);(9,12,14,26)(10,16,19,34)(11,21)(13,25,28,39)(15,29)(17,32,18,33)(20,31)(22,24)(23,36)(27,38)(30,37)(35,40)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=g,e^2=hi,f^2=i,b^a=bd,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=dhi,e^d=ehi,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57653);