Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #392 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
367 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 367 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) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,17)(12,20)(13,16)(14,18)(15,22)(19,24)(21,23);(2,6)(3,4)(7,8)(9,11)(10,14)(12,21)(13,19)(15,24)(16,23)(17,20)(18,22);(2,7)(3,4)(6,8)(14,23)(17,24)Polycyclic presentation:
2^8|d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94634, #94635, #125651, … |
| maximal normal subgroup of | #94634, #94635, #125651, … |
| maximal quotient of | #102541, #102548, #102551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3000);