Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
225 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 225 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) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,14,13,11,15,16)(17,18)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,14)(15,16)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(2,6)(5,7)(10,11)(14,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=gh,d^2=g,e^2=h,b^a=bd,c^a=cfg,d^a=df,c^b=ch,d^b=dg,e^b=efg46 conjugacy classes — view the full character table.
| maximal subgroup of | #98062, #98063, #98064, … |
| maximal normal subgroup of | #98062, #98063, #98064, … |
| maximal quotient of | #96295, #96756, #96757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 719);