Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) = C22×D4×B(4,𝔽2) (order 2048) · Inn(G) = #99 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C2×C4×D4 | C22 |
| socle | C22 | #342 |
| prime radical | C22×D4 | Q8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
162 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 41 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:
(2,5)(6,7)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24);(1,2)(3,7,8,5)(4,6)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=f,e^2=f,b^a=bc,c^a=cf,e^a=efg,c^b=cg,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7285, #127832, #127833, … |
| maximal normal subgroup of | C2×#7285, #127832, #127833, … |
| maximal quotient of | #93398, #93426, #93427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 273);