Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C2×#72225052 (order 4096) · Inn(G) = D16 · Out(G) = C2×C8×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D16 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | C4×C16 | C22 |
| socle, prime radical | C22 | #366 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(10,15)(11,20)(12,21)(13,24)(16,23)(17,32)(18,35)(19,29)(22,31)(25,38)(26,27)(28,33)(30,39)(34,40)(36,37);(9,10,16,30,12,17,22,28,24,33,37,32,38,39,11,15,14,19,20,27,25,34,36,35,13,18,31,40,21,26,23,29)Polycyclic presentation:
2^8|a^2=d,b^2=ceh,c^2=eg,d^2=f,e^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #94609, #94610, #94736, … |
| maximal normal subgroup of | #94609, #94610, #94736, … |
| maximal quotient of | #93881, #94021, #94752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 449);