Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = C2×#72236479 (order 2048) · Inn(G) = C2×D4 · Out(G) = C2×#525 |
| 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 | |
162 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 162 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:
(1,2)(3,4)(5,6,11,17,10,15,8,26,12,18,28,14,9,16,20,33)(7,19,24,34,23,31,27,32,25,30,35,36,22,13,21,29);(5,7,20,21,9,22,28,35,12,25,8,27,10,23,11,24)(6,13,26,34,16,30,17,29,18,31,33,36,15,19,14,32);(2,4)(5,8,9,11,12,20,10,28)(6,14,15,33,18,17,16,26)(7,21,22,35,25,27,23,24)(13,29,19,34,31,32,30,36)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=d,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13125, #494836, #494837, … |
| maximal normal subgroup of | C2×#13125, #494836, #494837, … |
| maximal quotient of | #140105, #140117, #140127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6113);