Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| 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 | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | #121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
180 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 180 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12)(3,13,4,15)(5,11,6,10)(8,16,9,14)(17,18,21,26,23,20,22,25)(19,27,30,28,31,32,29,24);(1,3,7,4)(2,8,12,9)(5,16,6,14)(10,15,11,13)(17,19,21,30,23,31,22,29)(18,24,26,27,20,28,25,32);(1,4,7,3)(2,9,12,8)(5,16,6,14)(10,15,11,13)(17,20,23,18)(19,28,31,24)(21,25,22,26)(27,30,32,29)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=eh,d^c=dh,e^c=eg46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8877, #192492, #192493, … |
| maximal normal subgroup of | C2×#8877, #192492, #192493, … |
| maximal quotient of | #99414, #99415, #99417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1865);