Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = C2×#72279669 (order 2048) · Inn(G) = C22⋊C4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #336 |
| prime radical | C23 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
133 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 133 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,13,18,16,17,23,24,15,20,22,21,12,11,14,19);(1,2,4,7,5,8,3,6)(9,11,14,21,12,20,22,24,15,17,23,18,16,10,13,19)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=efh,c^b=ch76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7169, #126763, #126800, … |
| maximal normal subgroup of | C2×#7169, #126763, #126800, … |
| maximal quotient of | #92947, #92948, #92949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 157);