Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8453 |
| automorphism | Aut(G) order 8192 |
| 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 | #3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
501 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,14,19,15,20,13,18)(11,21,25,16,24,31,23,30)(12,17,26,32,27,33,28,34)(22,36,37,40,38,35,39,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,25)(14,23)(15,24)(17,29)(18,31)(19,21)(20,30)(26,39)(27,38)(28,37)(32,35)(33,40)(34,36);(9,12,15,27)(10,17,20,33)(11,22,24,38)(13,28,14,26)(16,29,30,40)(18,34,19,32)(21,35,31,36)(23,39,25,37)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eh,f^b=fgh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93204, #94448, #94523, … |
| maximal normal subgroup of | #93204, #94448, #94523, … |
| maximal quotient of | #96965, #96966, #96967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1452);