Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10645 |
| automorphism | Aut(G) = C23×#45355 (order 2048) · Inn(G) = #397 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #397 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
129 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 129 conjugacy classes of subgroups — too many to draw. Only its 41 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:
(1,2,5,8,6,10,4,9)(3,7,11,16,13,14,12,15)(17,18,21,23,22,26,20,25)(19,27,29,24,30,31,28,32);(1,3)(2,7)(4,12)(5,11)(6,13)(8,16)(9,15)(10,14)(18,23)(19,28)(20,21)(25,26)(27,31)(29,30);(2,8)(3,11)(4,5)(7,14)(9,10)(12,13)(17,19,20,29,22,30,21,28)(18,24,25,31,26,32,23,27)Polycyclic presentation:
2^8|a^2=eh,c^2=d,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10647, #243093, #243094, … |
| maximal normal subgroup of | C2×#10647, #243093, #243094, … |
| maximal quotient of | #105215, #105217, #105218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3635);