Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) = #72306287 (order 4096) · Inn(G) = #102 · Out(G) = C2×#525 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | #466 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
158 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18,20,24,30,29,32,27,21,25,31,23,22,19,26,28);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,22)(20,24)(21,29)(23,31)(25,30)(26,27)(28,32);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^8|a^2=e,c^2=fh,d^2=gh,e^2=fg,g^2=h,b^a=bd,d^a=df,d^b=dg,e^b=efg,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #129005, #129006, #129023, … |
| maximal normal subgroup of | #129005, #129006, #129023, … |
| maximal quotient of | #124106, #124108, #124177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4932);