Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
308 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,13,15,16)(12,14)(17,18,20,23,21,19,22,24);(2,5)(6,7)(9,11,12,15)(10,13)(14,16)(17,19,20,24,21,18,22,23);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=f,b^2=defg,c^2=e,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dg,f^b=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #97679, #97681, #97685, … |
| maximal normal subgroup of | #97679, #97681, #97685, … |
| maximal quotient of | #102390, #102392, #102398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2867);