Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10378 |
| automorphism | Aut(G) = C23×#35346 (order 2048) · Inn(G) = #393 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| 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 | |
199 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 45 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:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,18,20,24)(12,16)(13,22)(14,17)(15,21,23,19);(3,8)(5,6)(11,19,20,21)(15,18,23,24);(2,5)(3,8)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,24)(17,23)(18,22)Polycyclic presentation:
2^8|a^2=d,b^2=d,e^2=gh,f^2=g,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,e^b=eh,d^c=dg,e^c=egh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10378, #238311, #238315, … |
| maximal normal subgroup of | C2×#10378, #238311, #238315, … |
| maximal quotient of | #104729, #104732, #104737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3366);