Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#394 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,18)(5,10)(6,11)(7,12)(8,16)(9,21)(14,25)(15,24)(17,28)(19,23)(20,27)(22,26)(29,31)(30,32);(1,3)(2,8)(4,19)(5,15)(6,16)(7,17)(9,25)(10,13)(11,28)(12,24)(14,22)(18,32)(20,31)(21,30)(23,27)(26,29);(1,4,5,20,7,22,6,21)(2,9,11,26,12,27,10,18)(3,14,16,31,17,19,15,30)(8,23,13,29,24,25,28,32);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,20,22,21)(8,13,24,28)(9,18,27,26)(14,30,19,31)(23,29,25,32)Polycyclic presentation:
2^8|c^2=e,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=cg,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #359161, #376294, #376488, … |
| maximal normal subgroup of | #359161, #376294, #376488, … |
| maximal quotient of | #238928, #238932, #238933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16513);