Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C25×#120 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| 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 | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
234 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 234 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)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(18,22,25,28)(19,26,27,29);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,18,23,22,21,25,20,28)(19,24,26,30,27,32,29,31);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,19)(18,24)(20,29)(21,27)(22,30)(23,26)(25,32)(28,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,d^2=g,e^2=g,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×#9038, #195064, #195066, … |
| maximal normal subgroup of | C2×#9038, #195064, #195066, … |
| maximal quotient of | #99583, #99584, #99585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2026);