Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7035 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #95 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C22×C42 | C22 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
245 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,13)(12,15)(14,16)(17,18)(19,21,23,24)(20,22);(2,6,8,7)(10,13)(11,16)(14,15)(17,19)(18,21,22,24)(20,23)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=fg,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7039, #95935, #95936, … |
| maximal normal subgroup of | C2×#7039, #95935, #95936, … |
| maximal quotient of | #92808, #92809, #92813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27);