Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C24 | C22×C4 |
| Frattini, prime radical, cosocle | C22×D4 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
244 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 244 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,16,12,15,18,23)(11,17,20,24,19,14,21,22);(2,5)(6,7)(9,11,18,20)(10,14,16,22)(12,19,13,21)(15,17,23,24);(1,3,4,8)(2,6,5,7)(10,15)(14,17)(16,23)(22,24)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=eh,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,f^b=fg,g^b=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132405, #132406, #132407, … |
| maximal normal subgroup of | #132405, #132406, #132407, … |
| maximal quotient of | #141936, #141951, #141962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6285);