Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | Aut(G) = C23×#2846 (order 1024) · Inn(G) = #379 · Out(G) = C24 |
| 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 | #3534 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
416 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 71 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)(9,10,13,17,14,18,12,22)(11,19,20,15,21,16,23,24);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,23)(13,20)(14,21)(16,17)(18,24)(19,22);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,20)(13,23)(14,21)(15,17)(18,19)(22,24)Polycyclic presentation:
2^8|a^2=f,d^2=h,f^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×#8487, #187296, #187297, … |
| maximal normal subgroup of | C2×#8487, #187296, #187297, … |
| maximal quotient of | #97164, #97166, #97167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1475);