Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13460 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #2058 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2058 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 59 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)(3,11,13,27)(4,16)(5,20)(6,9)(7,14,12,28)(8,18)(10,22)(15,29,21,24)(17,23,31,26)(19,30)(25,32);(1,2)(3,12,13,7)(4,8)(5,10)(6,9)(11,14,27,28)(15,26,21,23)(16,18)(17,29,31,24)(19,25)(20,22)(30,32);(1,3,5,21,6,13,22,15)(2,7,10,26,9,12,20,23)(4,17,18,14,19,31,32,28)(8,24,16,11,25,29,30,27)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=dg,e^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,f^b=fh,d^c=dh,e^c=eg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13460, #499212, #499269, … |
| maximal normal subgroup of | C2×#13460, #499212, #499269, … |
| maximal quotient of | #145798, #145799, #145828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6448);