Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C8 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #340 |
| prime radical | C23 | #103 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10,13,18,15,21,32,20)(11,22,24,35,27,37,26,40)(12,28,30,17,31,19,14,33)(16,25,34,23,36,39,38,29);(2,6,8,7)(9,11,14,25,32,24,31,23,15,27,30,39,13,26,12,29)(10,16,19,35,18,38,33,22,21,36,28,40,20,34,17,37)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fg,f^2=h,b^a=bc,c^a=cf,e^a=efh,f^a=fh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7177, #126783, #126844, … |
| maximal normal subgroup of | C2×#7177, #126783, #126844, … |
| maximal quotient of | #92941, #92945, #92957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 165);