Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
239 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 239 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)(4,12,15,10)(5,11)(7,9)(8,14,16,13)(17,18)(19,22,26,29)(20,23)(21,24)(25,28,31,32)(27,30);(1,3,6,2,7,11,5,9)(4,10,13,8,15,12,14,16)(17,19,21,26)(18,22)(20,25,27,31)(23,28)(24,29)(30,32);(1,4,7,15)(2,8,9,16)(3,10,11,12)(5,14,6,13)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=gh,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=egh,f^c=fgh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9242, #198112, #198114, … |
| maximal normal subgroup of | C2×#9242, #198112, #198114, … |
| maximal quotient of | #99888, #99889, #99896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2230);