Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11692 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 226 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,24,23);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14,15,20)(12,18)(13,17)(16,23,22,21)(19,24);(11,17)(14,20)(18,24)(21,23)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=gh,e^2=h,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=dg,e^b=efgh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #129807, #129808, #129823, … |
| maximal normal subgroup of | #129807, #129808, #129823, … |
| maximal quotient of | #124090, #124122, #124196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4710);