Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C24×#3484 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
212 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,5,7,6,10,4,9)(3,11,13,8,14,15,12,16)(17,18,21,25,22,19,23,27)(20,26,29,32,28,24,30,31);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19,21,27,22,18,23,25)(20,26,30,31,28,24,29,32);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,11,9,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,32)(27,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,e^a=egh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=eg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9048, #195354, #195356, … |
| maximal normal subgroup of | C2×#9048, #195354, #195356, … |
| maximal quotient of | #99580, #99586, #99588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2036);