Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
256 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 97 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(10,14,15,16)(17,18,20,22,21,24,19,23);(2,5)(6,7)(9,10,12,16,13,15,11,14)(18,22)(19,20)(23,24);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,14,15,16)(17,19,21,20)(18,23,24,22)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=eg,d^2=g,f^2=h,b^a=bd,c^a=ce,c^b=ch,d^b=dg,f^b=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #98040, #98044, #98046, … |
| maximal normal subgroup of | #98040, #98044, #98046, … |
| maximal quotient of | #102413, #102414, #102415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2856);