Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10374 |
| automorphism | Aut(G) = C23×#35346 (order 2048) · Inn(G) = #393 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
211 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8)(3,11,13,26)(5,19,17,9)(6,10,18,25)(7,22,23,30)(12,20,24,14)(15,28,16,29)(21,32,31,27);(1,2)(3,12,15,7)(4,8)(5,9)(6,10)(11,14,28,30)(13,24,16,23)(17,19)(18,25)(20,29,22,26)(21,27)(31,32);(1,3,4,16)(2,7,8,24)(5,20,17,30)(6,15,18,13)(9,26,19,28)(10,12,25,23)(11,27,29,32)(14,21,22,31)Polycyclic presentation:
2^8|a^2=d,b^2=dh,c^2=h,e^2=gh,b^a=bd,c^a=ce,e^a=eh,f^a=fg,c^b=cf,f^b=fg,d^c=dg,e^c=egh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10377, #238256, #238257, … |
| maximal normal subgroup of | C2×#10377, #238256, #238257, … |
| maximal quotient of | #104766, #104768, #104770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3365);