Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #121 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1939 |
| prime radical | C23 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 81 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:
(1,2,6,12,7,11,22,29,8,13,21,10,5,23,4,20)(3,14,18,31,16,9,24,27,19,26,32,30,17,28,15,25);(1,3,6,18,7,16,22,24,8,19,21,32,5,17,4,15)(2,9,12,27,11,26,29,30,13,28,10,25,23,14,20,31);(1,4,5,21,8,22,7,6)(2,10,11,20,13,12,23,29)(3,15,17,32,19,24,16,18)(9,25,26,31,28,27,14,30)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=e,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #135891, #135893, #135910, … |
| maximal normal subgroup of | #135891, #135893, #135910, … |
| maximal quotient of | #140337, #140340, #140352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6152);