Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| 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×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | #532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
178 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)(9,10,14,20,16,19,29,18,15,21,28,37,13,30,12,27)(11,22,25,38,24,17,32,34,26,36,40,33,31,35,23,39);(2,5)(3,7)(9,11)(10,17)(12,23)(13,31)(14,25)(15,26)(16,24)(18,33)(19,36)(20,34)(21,35)(22,30)(27,38)(28,40)(29,32)(37,39);(9,12,13,28,15,29,16,14)(10,18,30,20,21,27,19,37)(11,23,31,40,26,32,24,25)(17,33,22,34,35,38,36,39)Polycyclic presentation:
2^8|a^2=e,c^2=fg,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,d^b=dh,e^b=efh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94148, #94151, #128514, … |
| maximal normal subgroup of | #94148, #94151, #128514, … |
| maximal quotient of | #124202, #124211, #124238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4479);