Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| automorphism | Aut(G) = C4×#33553 (order 1024) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C8 |
| Frattini, cosocle | #408 | C22 |
| socle | C2 | #1316 |
| prime radical | #585 | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,12,25,42,8,13,26,43,6,27,30,41)(3,14,16,44,17,45,23,57,18,46,52,60,28,53,51,64)(4,19,20,54,21,10,36,37,22,38,24,58,29,39,56,63)(9,31,32,50,34,59,61,48,35,55,62,15,33,47,40,49);(1,3)(2,9)(4,15)(5,23)(6,28)(7,17)(8,18)(10,14)(11,40)(12,34)(13,35)(16,30)(19,53)(20,55)(21,49)(22,50)(24,31)(25,52)(26,51)(27,33)(29,48)(32,42)(36,47)(37,57)(38,45)(39,46)(41,62)(43,61)(44,54)(56,59)(58,60)(63,64)Polycyclic presentation:
2^8|a^2=d,d^2=f,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=deh,e^b=eh,g^b=gh,d^c=dgh,e^c=eh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7346, #149546, #149681, … |
| maximal normal subgroup of | C2×#7346, #149546, #149681, … |
| maximal quotient of | #93194, #93195, #93200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 334);