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)(11,16)(12,20)(13,23)(17,32)(18,35)(19,28)(21,37)(22,31)(24,36)(25,26)(27,33)(29,30)(34,40)(38,39);(9,10,16,26,12,17,29,35,23,33,37,40,36,38,22,28,14,19,31,39,24,34,21,27,13,18,30,32,20,25,11,15)Polycyclic presentation:
2^8|a^2=d,b^2=ce,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 | #94607, #94608, #94734, … |
| maximal normal subgroup of | #94607, #94608, #94734, … |
| maximal quotient of | #93880, #94021, #94749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 448);