Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13569 |
| automorphism | Aut(G) = #72286298 (order 2048) · Inn(G) = C2×AΣL(1,𝔽16) · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C4⋊Q8 | C23 |
| prime radical | C2×2−1+4 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,6,11,19,29)(3,12,15,23,13,25,31,7)(5,10,21,8,22,26,18,27)(14,30,20,28,17,24,16,32);(1,3)(2,7)(4,17)(5,20)(6,13)(8,25)(9,28)(10,30)(11,23)(12,27)(14,19)(15,18)(16,22)(21,31)(24,26)(29,32);(3,13)(7,23)(12,25)(14,17)(15,31)(16,20)(24,30)(28,32)Polycyclic presentation:
2^8|a^2=e,d^2=h,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=dh,e^b=ef,f^b=fh,g^b=gh,e^d=egh,f^d=fh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #152023, #152024, #152196, … |
| maximal normal subgroup of | #152023, #152024, #152196, … |
| maximal quotient of | #149543, #149647, #149680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6561);