Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C22×D4×#525 (order 2048) · Inn(G) = D4 · Out(G) = C22×#525 |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D4 |
| commutator | C4 | C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle, prime radical | C23 | C22⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
143 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,16,21,18,15,20,24,19,23,17,22,27,26,28,25);(2,4)(5,6,8,11,9,12,7,10)(13,15,17,25,18,23,28,21,19,26,16,24,27,14,20,22)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fg,g^2=h,b^a=bc,c^a=cf,e^a=ef112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7167, #126757, #126758, … |
| maximal normal subgroup of | C2×#7167, #126757, #126758, … |
| maximal quotient of | #92931, #92932, #92933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 155);