Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11799 |
| 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 | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(2,6)(3,4)(7,8)(9,10,12,16,13,15,20,23)(11,17,19,24,18,14,21,22);(1,2)(3,6)(4,8)(5,7)(9,11,13,18)(10,14,15,17)(12,19,20,21)(16,24,23,22);(1,3,5,4)(2,6,7,8)(11,18)(14,17)(19,21)(22,24)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=g,d^2=g,e^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,d^b=dg,e^b=efg,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #132247, #132248, #132263, … |
| maximal normal subgroup of | #132247, #132248, #132263, … |
| maximal quotient of | #124573, #124584, #124598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4822);