Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | #3492 | 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 | |
413 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,21,27,23,29,20,25)(19,30,31,45,33,46,47,42)(22,28,35,44,39,26,37,43)(24,36,40,38,41,34,48,32);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,24)(20,34)(21,36)(22,38)(23,33)(25,42)(26,30)(27,45)(28,46)(29,41)(31,35)(32,39)(37,47)(40,44)(43,48);(2,7)(6,14)(8,15)(9,16)(17,20,23,21)(18,25,29,27)(19,31,33,47)(22,37,39,35)(24,40,41,48)(26,44,28,43)(30,45,46,42)(32,36,38,34)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,e^d=ehi,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57936);