Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) = C4×#72224760 (order 6144) · Inn(G) = #127 · Out(G) = C2×C4×S4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1843 |
| prime radical | C23 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,6,11,21,32,7,12,22,33,23,34,47,58)(3,8,15,26,16,27,40,52,14,25,38,50,39,51,60,63)(4,9,19,30,20,31,46,57,18,29,44,55,45,56,62,64)(13,35,37,59,17,41,43,61,36,24,48,49,42,28,53,54);(1,3)(2,8)(4,17,18,42)(5,15)(6,16)(7,14)(9,28,29,41)(10,26)(11,27)(12,25)(13,20,36,45)(19,43,44,53)(21,40)(22,38)(23,39)(24,31,35,56)(30,54,55,61)(32,52)(33,50)(34,51)(37,46,48,62)(47,60)(49,57,59,64)(58,63);(1,4)(2,9)(3,13,14,36)(5,19)(6,20)(7,18)(8,24,25,35)(10,30)(11,31)(12,29)(15,37,38,48)(16,17,39,42)(21,46)(22,44)(23,45)(26,49,50,59)(27,28,51,41)(32,57)(33,55)(34,56)(40,43,60,53)(47,62)(52,54,63,61)(58,64)Polycyclic presentation:
2^8|a^2=f,b^2=d,c^2=e,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=cg,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #12902⋊C3 |
| maximal subgroup of | #93741, #93742, #145496, … |
| maximal normal subgroup of | #93741, #93742, #145496, … |
| maximal quotient of | #135741, #135746, #135752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5890);