Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3482 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,7)(5,9)(6,14)(8,12)(11,15)(13,16)(17,18,21,24,22,26,20,25)(19,27,29,23,30,31,28,32);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(17,20,22,21)(18,25,26,24)(19,28,30,29)(23,27,32,31)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eg,d^c=dgh58 conjugacy classes — view the full character table.
| maximal subgroup of | #221124, #221128, #221132, … |
| maximal normal subgroup of | #221124, #221128, #221132, … |
| maximal quotient of | #164371, #164374, #164377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10511);