Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
222 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6,8,11)(3,9,10,12)(17,18,21,26,22,24,20,25)(19,27,30,23,29,32,28,31);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,19)(18,23)(20,30)(21,28)(22,29)(24,31)(25,32)(26,27);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(18,24)(20,21)(23,31)(28,30)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #153301, #153302, C2×#9067, … |
| maximal normal subgroup of | #153301, #153302, C2×#9067, … |
| maximal quotient of | #99595, #99598, #99605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2055);