Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
213 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 213 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,11,17,12,19,28,18)(7,14,22,34,23,31,36,35)(8,24,20,13,26,32,21,33)(9,27,29,15,10,30,25,16);(1,2,3,4)(5,7,10,21,28,36,29,26,12,23,9,20,11,22,25,8)(6,13,15,14,17,32,30,34,19,33,16,31,18,24,27,35);(5,8,25,22,11,20,9,23,12,26,29,36,28,21,10,7)(6,14,30,33,18,35,15,32,19,31,27,13,17,34,16,24)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #93917, #93918, #152525, … |
| maximal normal subgroup of | #93917, #93918, #152525, … |
| maximal quotient of | #140722, #140723, #140724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6229);