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 | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
226 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 226 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)(3,6,10,13)(4,7)(5,8)(9,12,15,16)(11,14)(18,22,24,27)(19,25,26,28);(1,3)(2,6,8,13)(4,9)(5,10)(7,12,14,16)(11,15)(17,18,20,27,21,24,29,22)(19,23,28,31,26,32,25,30);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,19)(18,23)(20,28)(21,26)(22,30)(24,32)(25,29)(27,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,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×#9240, #198112, #198113, … |
| maximal normal subgroup of | C2×#9240, #198112, #198113, … |
| maximal quotient of | #99884, #99885, #99892, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2228);