Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | #532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10,14,20,16,19,29,37,15,21,28,38,13,30,12,18)(11,22,25,36,24,17,23,33,26,35,39,32,31,34,27,40);(2,5)(3,7)(9,11)(10,17)(12,27)(13,31)(14,25)(15,26)(16,24)(18,36)(19,35)(20,33)(21,34)(22,30)(23,29)(28,39)(32,37)(38,40);(9,12,13,28,15,29,16,14)(10,18,30,38,21,37,19,20)(11,23,31,25,26,27,24,39)(17,32,22,33,34,36,35,40)Polycyclic presentation:
2^8|a^2=e,c^2=fg,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=df,c^b=ch,d^b=dh,e^b=efh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129662, #129663, #129664, … |
| maximal normal subgroup of | #129662, #129663, #129664, … |
| maximal quotient of | #124203, #124212, #124239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4480);