Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8453 |
| automorphism | Aut(G) = C22×#72213827 (order 4096) · Inn(G) = #379 · Out(G) = C23×D4 |
| 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 | #3492 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
419 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 419 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) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(3,8)(5,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^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 | #93192, #140018, #140019, … |
| maximal normal subgroup of | #93192, #140018, #140019, … |
| maximal quotient of | #96964, #96968, #96969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1442);