Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | Aut(G) = C23×#7684106 (order 4096) · Inn(G) = #121 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | C23 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
237 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 71 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:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,23,20,28)(21,26)(22,25)(24,29,32,27)(30,31);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19)(18,23,25,28)(20,22)(21,29)(24,26,32,31)(27,30);(3,11)(7,15)(10,16)(12,13)(17,20,22,19)(18,24,25,32)(21,27,30,29)(23,31,28,26)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9236, #198021, #198024, … |
| maximal normal subgroup of | C2×#9236, #198021, #198024, … |
| maximal quotient of | #99860, #99862, #99868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2224);