Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10508 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #394 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3213 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
207 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 47 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)(4,7)(6,8)(9,10)(11,21)(12,16)(13,19)(14,18)(15,20)(17,23)(22,24);(3,8)(5,6)(9,11)(10,16)(12,14)(13,23)(15,22)(17,20)(18,24)(19,21);(2,5)(3,8)(6,7)(9,12,13,22,15,23,14,11)(10,17,18,24,20,16,19,21)Polycyclic presentation:
2^8|c^2=d,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=egh,f^b=fg,e^c=egh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10508, #240861, #240865, … |
| maximal normal subgroup of | C2×#10508, #240861, #240865, … |
| maximal quotient of | #104897, #104898, #104899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3496);