Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10500 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #394 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
214 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(2,7)(3,8)(4,5)(9,10)(11,14)(12,15)(13,17)(16,21)(18,22)(19,24)(20,23);(1,2)(3,4)(5,7)(6,8)(9,11,13,19)(10,14,17,24)(12,20,21,18)(15,23,16,22);(1,3,4,2,6,7,5,8)(10,15)(12,21)(14,22)(16,17)(18,20)(23,24)Polycyclic presentation:
2^8|b^2=gh,c^2=d,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,e^b=eh,f^b=fg,e^c=egh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10503, #240748, #240750, … |
| maximal normal subgroup of | C2×#10503, #240748, #240750, … |
| maximal quotient of | #104927, #104928, #104935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3491);