Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) = C2×#72237540 (order 2048) · Inn(G) = #124 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1985 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
158 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,6,12,7,11,23,26,8,13,24,30,5,19,15,31)(3,14,18,22,17,27,32,20,16,9,25,21,10,28,29,4);(1,3,15,29,5,10,24,25,8,16,23,32,7,17,6,18)(2,9,26,4,19,27,12,21,13,14,31,20,11,28,30,22);(1,4,5,21,8,20,7,22)(2,10,11,3,13,17,19,16)(6,14,15,28,24,9,23,27)(12,29,26,18,30,32,31,25)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cg,d^b=dh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94479, #94480, #94481, … |
| maximal normal subgroup of | #94479, #94480, #94481, … |
| maximal quotient of | #141865, #141866, #141870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6258);