Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #379 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | #3536 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
414 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 73 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)(3,4,7,8)(5,6)(9,10)(11,18,20,15)(12,14)(13,17)(16,21)(19,23,24,22);(1,3)(2,4)(5,7)(6,8)(10,14)(12,21)(15,22)(16,17)(18,23)(19,24);(1,4)(2,3)(5,8)(6,7)(9,11,13,20)(10,15,17,18)(12,19,21,24)(14,22,16,23)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,f^c=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8480, #187155, #187156, … |
| maximal normal subgroup of | C2×#8480, #187155, #187156, … |
| maximal quotient of | #97175, #97177, #97181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1468);