Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | Aut(G) = C22×#72231762 (order 4096) · Inn(G) = #386 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
222 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,11,12,15,14,16,13,7)(17,18,21,25,22,26,20,24)(19,27,29,23,30,31,28,32);(1,3)(2,7,10,15)(4,13)(5,12)(6,14)(8,16,9,11)(17,19)(18,23)(20,29)(21,28)(22,30)(24,31)(25,27)(26,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9328, #199431, #199435, … |
| maximal normal subgroup of | C2×#9328, #199431, #199435, … |
| maximal quotient of | #99972, #99974, #99980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2316);