Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) = #72225052 (order 2048) · Inn(G) = C2×D16 · Out(G) = C4×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | C4×D16 |
| prime radical | #2262 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,8,15,30,14)(3,16,19,47,21,48,20,54)(4,13,23,39,24,10,7,31)(5,25,27,57,29,38,62,58)(9,33,35,53,37,52,45,18)(11,22,41,55,43,56,42,28)(17,49,50,64,51,46,32,40)(26,44,59,63,61,34,60,36);(1,3)(2,9)(4,17)(5,26)(6,19)(7,32)(8,21)(10,34)(11,40)(12,35)(13,44)(14,45)(15,37)(16,25)(18,28)(20,30)(22,33)(23,50)(24,51)(27,59)(29,61)(31,36)(38,48)(39,63)(41,49)(42,46)(43,64)(47,57)(52,56)(53,55)(54,58)(60,62);(1,4,6,23,8,24,30,7)(2,10,12,31,15,13,14,39)(3,17,19,50,21,51,20,32)(5,22,27,55,29,56,62,28)(9,34,35,36,37,44,45,63)(11,38,41,58,43,25,42,57)(16,46,47,40,48,49,54,64)(18,26,33,59,53,61,52,60)Polycyclic presentation:
2^8|a^2=e,c^2=e,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94613, #151086, #153597, … |
| maximal normal subgroup of | #94613, #151086, #153597, … |
| maximal quotient of | #151078, #151079, #151080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6644);