Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C22 | C82 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #336 |
| prime radical | C23 | C4×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(9,10,12,16,13,18,23,30,14,17,24,29,25,31,36,39)(11,19,21,32,22,33,35,40,20,15,26,27,34,28,37,38);(1,2,3,5,4,6,7,8)(9,11)(10,15,17,19)(12,21)(13,22)(14,20)(16,27,29,32)(18,28,31,33)(23,35)(24,26)(25,34)(30,38,39,40)(36,37)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,b^a=bc,e^a=eh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7140, #126557, #126568, … |
| maximal normal subgroup of | C2×#7140, #126557, #126568, … |
| maximal quotient of | #92898, #92899, #92901, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 128);