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) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
202 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 202 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7)(3,5,8,6)(10,15)(11,16)(12,13)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28);(9,10,14,11)(12,16,13,15)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,28,30,31);(2,5)(3,8)(6,7)(9,11,14,10)(12,16,13,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,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×#7828, #165748, #165761, … |
| maximal normal subgroup of | C2×#7828, #165748, #165761, … |
| maximal quotient of | #97541, #99366, #99371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 816);