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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| prime radical | C2×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
246 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 246 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)(3,11,14,29)(4,16,18,28)(5,20,22,9)(6,10)(7,13,24,31)(8,19)(12,26,15,30)(17,25,32,23)(21,27);(1,3)(2,7)(4,17)(5,14)(6,12)(8,26)(9,24)(10,23)(11,27)(13,18)(15,22)(16,30)(19,31)(20,25)(21,32)(28,29);(3,12)(7,23)(11,26)(13,17)(14,15)(24,25)(29,30)(31,32)Polycyclic presentation:
2^8|a^2=e,d^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 | #152021, #152022, #152179, … |
| maximal normal subgroup of | #152021, #152022, #152179, … |
| maximal quotient of | #149543, #149647, #149679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6557);