Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #99 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C22×C4⋊C4 | C22 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
231 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6,7,9,8,10,11,12)(13,14)(15,18,20,17)(16,19);(1,2,3,4)(14,17,19,18)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=g,b^a=bc,c^a=cf,e^a=eh,f^a=fh,d^b=dfh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93895, #93896, #93905, … |
| maximal normal subgroup of | #93895, #93896, #93905, … |
| maximal quotient of | #92967, #92969, #92975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 177);