Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #476 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1920 |
| prime radical | C23 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
127 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 127 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:
(1,2,4,8,7,12,6,10)(3,13,16,27,17,30,14,29)(5,19,18,28,22,26,21,11)(9,20,24,15,25,31,23,32);(2,8,12,10)(9,23,25,24)(11,19,28,26)(13,29,30,27);(1,3,4,14,7,17,6,16)(2,9,10,24,12,25,8,23)(5,20,18,32,22,31,21,15)(11,27,26,30,28,29,19,13)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=d,d^2=h,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^c=eg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13117, #494803, #494999, … |
| maximal normal subgroup of | C2×#13117, #494803, #494999, … |
| maximal quotient of | #140079, #140096, #146887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6105);