Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #357 |
| commutator | C16 | C42 |
| Frattini, cosocle | C22×C16 | C22 |
| prime radical | C23 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
87 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(6,12)(7,17)(8,21)(9,19)(10,20)(13,14)(15,32)(16,18)(22,35)(23,33)(24,34)(25,26)(27,31)(28,30)(29,36);(5,6,8,13,23,28,7,18,20,31,35,26,19,25,34,27,11,16,24,30,9,14,22,12,10,15,17,29,33,36,21,32)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,e^2=fh,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7478, #151778, C4×#1419, … |
| maximal normal subgroup of | C2×#7478, #151778, C4×#1419, … |
| maximal quotient of | #94026, #94049, #94051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 466);