Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,24,27,36,44,23,29,35,45,21,37,20,26)(19,30,33,40,32,25,39,41,34,43,47,48,38,42,31,46);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,34)(18,25,29,42)(20,31,36,39)(21,38,24,32)(22,33,35,47)(26,40,44,48)(27,43,37,30)(28,41,45,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,35,23,36,24,22)(18,26,37,45,29,44,27,28)(19,31,38,47,34,39,32,33)(25,40,30,46,42,48,43,41)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=fg,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=df,d^b=dh,e^b=efh76 conjugacy classes — view the full character table.
| maximal subgroup of | #128510, #128511, #128512, … |
| maximal normal subgroup of | #128510, #128511, #128512, … |
| maximal quotient of | #124207, #124225, #124413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4484);