Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22⋊C4 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C22×C42 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
311 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,19)(16,18)(17,20);(6,9,11,10)(13,15)(14,17,18,19)(16,20)(21,22,23,25,24,26,27,28)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,e^2=h,b^a=bc,c^a=cf,e^a=efg,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | #93905, #93951, #93955, … |
| maximal normal subgroup of | #93905, #93951, #93955, … |
| maximal quotient of | #92808, #92813, #92814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13);