Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | Aut(G) = #72286229 (order 2048) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C4⋊Q8 | C23 |
| socle | C2 | C2×AΣL(1,𝔽16) |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
241 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 241 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:
(1,2,5,11,7,13,21,29)(3,14,17,25,18,27,32,8)(4,9,19,26,6,12,23,10)(15,31,16,30,22,28,20,24);(1,3)(2,8)(4,15)(5,20)(6,22)(7,18)(9,24)(10,27)(11,28)(12,30)(13,25)(14,26)(16,21)(17,19)(23,32)(29,31);(1,4,7,6)(2,9,13,12)(3,15,18,22)(5,19,21,23)(8,24,25,30)(10,11,26,29)(14,31,27,28)(16,32,20,17)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=dh,e^b=ef,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 | #151915, #151916, #152025, … |
| maximal normal subgroup of | #151915, #151916, #152025, … |
| maximal quotient of | #149542, #149646, #149677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6560);