Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10560 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #394 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | #3192 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
245 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 245 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)(4,6)(7,8)(10,15)(11,18)(12,13)(16,23)(17,21)(19,20);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,21)(13,15)(14,17)(18,22)(19,24)(20,23);(5,8)(9,11,12,19,14,20,13,18)(10,16,15,22,17,23,21,24)Polycyclic presentation:
2^8|c^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fh,c^b=cf,d^b=dg,e^b=eh,f^c=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10560, #241055, #241056, … |
| maximal normal subgroup of | C2×#10560, #241055, #241056, … |
| maximal quotient of | #104877, #104878, #104883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3548);