Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10212 |
| automorphism | Aut(G) = C22×#72231160 (order 4096) · Inn(G) = #392 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3214 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
251 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 49 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)(3,5)(4,7)(6,8)(11,18)(12,13)(15,19)(16,22)(20,21)(23,24);(2,5)(3,8)(6,7)(9,10)(11,19)(12,16)(13,22)(14,17)(15,21)(18,23)(20,24);(1,2,4,7)(3,6,8,5)(9,11)(10,15)(12,21)(13,18)(14,20)(16,24)(17,23)(19,22)Polycyclic presentation:
2^8|c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,f^b=fgh,d^c=dh,e^c=eh,f^c=fgh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10214, #232856, #232859, … |
| maximal normal subgroup of | C2×#10214, #232856, #232859, … |
| maximal quotient of | #102507, #102513, #102515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3202);