Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150578 |
| 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 | #391 | C23 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
603 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 603 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)(3,5)(4,6)(7,8)(9,10,13,19,15,20,30,35)(11,21,24,33,26,36,40,16)(12,27,28,37,14,17,31,18)(22,38,29,32,25,39,23,34);(2,5)(3,8)(6,7)(9,11)(10,16)(12,25)(13,29)(14,22)(15,26)(17,34)(18,36)(19,38)(20,33)(21,37)(23,30)(24,28)(27,32)(31,40)(35,39);(2,6)(5,7)(9,12,15,14)(10,17,20,27)(11,22,26,25)(13,28,30,31)(16,32,33,34)(18,35,37,19)(21,39,36,38)(23,24,29,40)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=ci,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^d=ehi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72340662, #72340683, #72340686 |
| maximal normal subgroup of | #72340662, #72340683, #72340686 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57779);