Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| 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 | #3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
502 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 77 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)(3,7,8,5)(4,6)(10,14,16,19)(15,23,24,22);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,19)(13,16)(14,21)(17,22)(18,24)(20,23);(1,3)(2,5)(4,8)(6,7)(9,11,13,18)(10,15,16,24)(12,20,21,17)(14,22,19,23)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=eh,f^b=fgh,d^c=dh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8467, #186963, #186964, … |
| maximal normal subgroup of | C2×#8467, #186963, #186964, … |
| maximal quotient of | #97318, #97320, #97324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1455);