Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | Aut(G) = C27×#120 (order 4096) · Inn(G) = #120 · Out(G) = C27 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
175 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 175 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) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,4)(7,8)(10,13)(11,14)(15,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28);(1,2,5,7)(3,6,4,8)(9,10,12,13)(11,14,15,16)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,28,30,31);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=f,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,d^b=df,e^b=eg,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7806, #165786, #165792, … |
| maximal normal subgroup of | C2×#7806, #165786, #165792, … |
| maximal quotient of | #97544, #99374, #99381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 794);