Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #382 | C23 |
| prime radical | #24223 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 447 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:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19,20,38,23,35,22,37)(12,24,26,36)(13,18)(14,16)(15,21,31,39,32,25,30,40)(17,27,33,28)(29,34);(2,6)(3,4)(7,8)(9,11)(10,15)(12,25)(13,20)(14,23)(16,32)(17,35)(18,31)(19,33)(21,26)(22,29)(24,38)(27,39)(28,40)(30,34)(36,37);(1,3,5,4)(2,7,8,6)(10,16)(11,20,23,22)(12,26)(15,30,32,31)(18,34)(19,37,35,38)(21,39,25,40)(27,28)Polycyclic presentation:
2^9|a^2=e,c^2=ghi,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ci,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=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57705);