Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | Aut(G) = C27×#120 (order 4096) · Inn(G) = #120 · Out(G) = C27 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
285 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3)(5,6,9,12)(7,10,8,11)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,17)(16,19)(18,20);(1,3,4,2)(6,11)(7,8)(10,12)(17,20)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=d,e^2=g,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,e^c=eg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10400, #239407, #239411, … |
| maximal normal subgroup of | C2×#10400, #239407, #239411, … |
| maximal quotient of | #104832, #104834, #104843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3388);