Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3217 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
240 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,4,6,5,3,8,7)(9,10,15,18)(11,14,17,13)(12,19)(16,20)(21,23)(22,24);(1,3)(2,5)(4,6)(7,8)(9,11,13,10,15,17,14,18)(12,19,21,16,22,24,20,23);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,24)(18,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,d^2=gh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=egh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9340, #199641, #199643, … |
| maximal normal subgroup of | C2×#9340, #199641, #199643, … |
| maximal quotient of | #100001, #100002, #100009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2328);