Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#107 |
| prime radical | C23 | #107 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,20)(12,17)(13,27)(14,29)(15,19)(16,23)(18,21)(22,33)(24,39)(25,35)(26,32)(28,40)(30,37)(31,38)(34,36);(9,11,21,36,14,24,28,38,15,25,26,37,12,22,13,23)(10,16,27,33,17,30,32,35,19,31,40,39,29,34,18,20);(2,5)(6,8)(9,12,15,14)(10,17,19,29)(11,22,25,24)(13,26,28,21)(16,30,31,34)(18,27,32,40)(20,33,35,39)(23,37,38,36)Polycyclic presentation:
2^8|a^2=e,b^2=df,c^2=h,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #94203, #94205, #94207, … |
| maximal normal subgroup of | #94203, #94205, #94207, … |
| maximal quotient of | #126686, #129112, #129113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5012);