Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12093 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | C25 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 37 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:
(1,2,5,9)(3,11,14,22)(4,15,18,8)(6,10,19,24)(7,16,21,13)(12,25,27,32)(17,28,30,23)(20,29,31,26);(1,3,5,14)(2,7,9,21)(4,16,18,13)(6,12,19,27)(8,22,15,11)(10,20,24,31)(17,29,30,26)(23,32,28,25);(3,12)(4,17)(7,20)(11,25)(14,27)(18,30)(21,31)(22,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,c^b=ch,d^b=df,e^b=egh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | #151391, #151462, C2×#12217, … |
| maximal normal subgroup of | #151391, #151462, C2×#12217, … |
| maximal quotient of | #125972, #126013, #126083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5205);