Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12148 |
| automorphism | Aut(G) = C25×B(4,𝔽2) (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | #392 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
219 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 219 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:
(1,2,5,10)(3,12,15,29)(4,7,16,9)(6,11,19,23)(8,22,25,14)(13,27,30,28)(17,20,32,24)(18,21,31,26);(2,7)(4,16)(8,20)(9,10)(11,21)(12,27)(13,30)(17,32)(18,31)(23,26)(24,25)(28,29);(1,3,5,15)(2,8,23,29)(4,17,16,32)(6,14,19,22)(7,20,26,28)(9,24,21,27)(10,25,11,12)(13,31,30,18)Polycyclic presentation:
2^8|a^2=f,c^2=fgh,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,d^b=df,e^b=egh,d^c=dg,e^c=eh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12263, #484822, #484829, … |
| maximal normal subgroup of | C2×#12263, #484822, #484829, … |
| maximal quotient of | #125920, #125951, #126004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5251);