Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150198 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | C2×#537 | C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
653 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 653 conjugacy classes of subgroups — too many to draw. Only its 74 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)(4,7)(6,8)(9,10,12,17,14,19,26,35)(11,20,22,33,21,36,37,40)(13,18,29,34,28,16,25,32)(15,24,31,27,30,38,39,23);(2,5)(3,8)(6,7)(9,11)(10,15)(12,24)(13,27)(14,21)(16,20)(17,33)(18,36)(19,30)(22,29)(23,28)(25,37)(26,38)(31,34)(32,39)(35,40);(11,21)(15,30)(20,36)(22,37)(23,27)(24,38)(31,39)(33,40)Polycyclic presentation:
2^9|a^2=e,d^2=g,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^d=ehi,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72340662, #72340679, #72340681 |
| maximal normal subgroup of | #72340662, #72340679, #72340681 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57398);