Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12436 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1667 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,9,16)(3,17,23,47)(4,24,27,55)(5,13,29,41)(6,12,28,42)(7,15,8,14)(10,32,37,19)(11,38,40,25)(18,48,51,59)(20,36,52,35)(21,46,22,34)(26,56,31,43)(30,44,57,39)(33,50,60,54)(45,58,62,49)(53,63,64,61);(1,3,6,20,7,21,29,32,9,23,28,52,8,22,5,19)(2,10,13,34,14,35,42,47,16,37,41,46,15,36,12,17)(4,18,30,53,26,50,38,58,27,51,57,64,31,54,25,49)(11,33,43,61,39,59,55,62,40,60,56,63,44,48,24,45);(1,4)(2,11)(3,18)(5,25)(6,30)(7,26)(8,31)(9,27)(10,33)(12,24)(13,43)(14,39)(15,44)(16,40)(17,45)(19,49)(20,53)(21,50)(22,54)(23,51)(28,57)(29,38)(32,58)(34,61)(35,59)(36,48)(37,60)(41,56)(42,55)(46,63)(47,62)(52,64)Polycyclic presentation:
2^8|a^2=h,b^2=df,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh,e^c=eg,g^c=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151526, #151527, #151529, … |
| maximal normal subgroup of | #151526, #151527, #151529, … |
| maximal quotient of | #123365, #123366, #123369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5425);