Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) = C25×#120 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | #121 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
169 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 169 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,6,10,7,11,5,13)(3,12,9,8,15,16,14,4)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,3,6,9,7,15,5,14)(2,8,10,16,11,4,13,12)(20,29)(23,28)(25,32)(27,30);(1,4)(2,9)(3,13)(5,16)(6,12)(7,8)(10,15)(11,14)(17,19)(18,22)(20,30)(21,26)(23,32)(24,31)(25,28)(27,29)Polycyclic presentation:
2^8|a^2=f,b^2=fh,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dg,e^b=eh,d^c=dh,e^c=egh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8917, #193108, #193110, … |
| maximal normal subgroup of | C2×#8917, #193108, #193110, … |
| maximal quotient of | #99326, #99327, #99329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1905);