Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12152 |
| automorphism | Aut(G) = C25×B(4,𝔽2) (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
198 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12)(3,8,14,10)(4,9,18,26)(6,22,7,13)(11,20,29,21)(15,27,17,25)(16,32,30,28)(19,24,31,23);(2,8)(3,14)(4,18)(9,23)(10,12)(11,28)(13,25)(15,17)(20,21)(22,27)(24,26)(29,32);(2,9,13,11)(3,15)(8,23,25,28)(10,24,27,32)(12,26,22,29)(14,17)(16,31)(19,30)Polycyclic presentation:
2^8|a^2=f,c^2=fgh,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gh,d^b=df,e^b=egh,d^c=dg,e^d=eh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12271, #484850, #484857, … |
| maximal normal subgroup of | C2×#12271, #484850, #484857, … |
| maximal quotient of | #125920, #125951, #126003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5259);