Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9784 |
| automorphism | Aut(G) = C22×#72232559 (order 4096) · Inn(G) = #390 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
255 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 65 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)(3,5,7,8)(4,6)(9,10,13,17,14,18,12,15)(11,19,22,24,20,23,21,16);(1,3)(2,5,6,8)(4,7)(10,15,18,17)(11,20)(16,19,24,23)(21,22);(3,7)(5,8)(9,11,12,21,14,20,13,22)(10,16,17,19,18,24,15,23)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=e,d^2=gh,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=cg,d^b=dh,f^b=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9816, #206022, #206027, … |
| maximal normal subgroup of | C2×#9816, #206022, #206027, … |
| maximal quotient of | #102069, #102070, #102071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2804);