Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) = C27×#442 (order 8192) · Inn(G) = C2×D8 · Out(G) = C26×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 139 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,23,26,30)(21,24,28,29)(25,31,32,27);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,20,26)(18,23,22,30)(21,25,28,32)(24,31,29,27);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,22)(19,25)(21,29)(23,27)(24,28)(26,32)(30,31)Polycyclic presentation:
2^8|a^2=dh,b^2=de,c^2=d,f^2=gh,g^2=h,b^a=bd,c^a=ce,c^b=cf,f^b=fg,g^b=gh,f^c=fg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13539, #499551, #499553, … |
| maximal normal subgroup of | C2×#13539, #499551, #499553, … |
| maximal quotient of | #146285, #148706, #148981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6527);