Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator, socle | C22 | C23×C8 |
| prime radical | #532 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,16,25,32,9,14,23,30,24,31,27,36)(7,17,19,33,20,21,29,35,18,12,26,28,34,8,13,22);(2,4)(5,7)(6,12)(8,16)(9,18)(10,19)(11,20)(13,27)(14,17)(15,28)(21,31)(22,32)(23,26)(24,34)(25,29)(30,33)(35,36);(5,8,10,22,11,17,25,33,9,21,23,35,24,12,27,28)(6,13,15,7,16,19,32,20,14,29,30,18,31,26,36,34);(7,18)(8,21)(12,17)(13,29)(19,26)(20,34)(22,35)(28,33)Polycyclic presentation:
2^8|a^2=f,c^2=f,f^2=g,g^2=h,b^a=be,d^b=dh,d^c=dh100 conjugacy classes — view the full character table.
| maximal subgroup of | #128703, #128718, #134945, … |
| maximal normal subgroup of | #128703, #128718, #134945, … |
| maximal quotient of | #483491, #483526, #483537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26426);