Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) = #72278809 (order 2048) · Inn(G) = #99 · Out(G) = C22×M16 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C22×C4 | C2×C8 |
| Frattini, cosocle | C2×C4⋊C8 | C22 |
| socle | C22 | #324 |
| prime radical | C23 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,9,11,15)(4,12,13,7)(5,8)(6,10)(14,16)(17,18,20,25,21,26,31,39,22,24,30,36,32,37,43,46)(19,23,28,33,29,35,42,44,27,34,40,38,41,45,48,47);(1,3)(2,6,8,14)(4,10)(5,11)(7,9,12,15)(13,16)(17,19)(18,23,24,34)(20,28)(21,29)(22,27)(25,38,36,33)(26,35,37,45)(30,40)(31,42)(32,41)(39,47,46,44)(43,48)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=gh,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=ch,d^b=degh,d^c=dg46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7065, #122328, #122329, … |
| maximal normal subgroup of | C2×#7065, #122328, #122329, … |
| maximal quotient of | #92871, #92991, #93736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53);