Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| automorphism | Aut(G) = #72286229 (order 2048) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | #127 | C23 |
| socle | C2 | C2×AΣL(1,𝔽16) |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
243 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 243 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)(3,12,14,7)(4,8)(5,10,19,9)(6,11)(13,26,27,20)(15,25,18,21)(16,23,30,22)(17,24)(28,32,29,31);(1,3)(2,7)(4,13)(5,18)(6,15)(8,20)(9,12)(10,25)(11,21)(14,19)(16,29)(17,28)(22,26)(23,32)(24,31)(27,30);(1,4,6,17)(2,8,11,24)(3,13,15,28)(5,16,19,30)(7,20,21,31)(9,22,10,23)(12,26,25,32)(14,27,18,29)Polycyclic presentation:
2^8|a^2=e,c^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,e^b=efh,f^b=fh,g^b=gh,e^d=egh,f^d=fh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151909, #151910, #152025, … |
| maximal normal subgroup of | #151909, #151910, #152025, … |
| maximal quotient of | #149538, #149642, #149674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6548);