Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C25×C4×D4 (order 1024) · Inn(G) = C2×D4 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C8×D4 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
163 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,11,16,12,15,30,33,9,14,18,19,10,27,29,35)(7,17,22,34,23,28,32,36,20,8,24,26,21,25,13,31);(1,2,4,3)(5,7,18,24,12,23,29,13,9,20,11,22,10,21,30,32)(6,8,19,34,15,25,35,36,14,17,16,26,27,28,33,31);(1,3,4,2)(5,8,9,17)(6,13,14,32)(7,19,20,16)(10,28,12,25)(11,26,18,34)(15,22,27,24)(21,33,23,35)(29,36,30,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=d,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,e^c=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13122, #494829, #494830, … |
| maximal normal subgroup of | C2×#13122, #494829, #494830, … |
| maximal quotient of | #140102, #140103, #140114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6110);