Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12096 |
| automorphism | Aut(G) order 4096 |
| 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 | |
196 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 196 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:
(1,2,6,13)(3,15,18,32)(4,21,23,31)(5,25,16,11)(7,27,8,14)(9,24,30,17)(10,26,29,22)(12,19,28,20);(1,3,7,19)(2,9,27,21)(4,16,17,26)(5,24,22,23)(6,18,8,20)(10,28,25,15)(11,32,29,12)(13,30,14,31);(1,4)(2,10)(3,16)(5,18)(6,23)(7,17)(8,24)(9,28)(11,14)(12,30)(13,29)(15,21)(19,26)(20,22)(25,27)(31,32)Polycyclic presentation:
2^8|a^2=f,b^2=fh,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^c=eh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151385, #151386, #151412, … |
| maximal normal subgroup of | #151385, #151386, #151412, … |
| maximal quotient of | #125970, #126012, #126083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5209);