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) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,3,7,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23)(25,26)(27,33,35,40)(28,32)(29,31)(30,34,38,36)(37,39);(1,3)(2,5)(4,6)(7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27)(26,30)(28,36)(29,35)(31,38)(32,40)(33,39)(34,37);(2,3)(6,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,31)(30,38)(32,39)(33,40)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=f,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 | #97361, #97362, #97364, … |
| maximal normal subgroup of | #97361, #97362, #97364, … |
| maximal quotient of | #96292, #96756, #96757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 705);