Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11799 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
194 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,18,14,17,26,33)(11,19,21,34,22,15,27,29)(12,16,24,32,25,31,37,40)(20,30,35,39,23,28,36,38);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,17,19)(12,23,25,20)(13,21,26,27)(16,30,31,28)(18,34,33,29)(24,36,37,35)(32,38,40,39);(2,6)(5,7)(9,12,14,25)(10,16,17,31)(11,20,22,23)(13,24,26,37)(15,28,19,30)(18,32,33,40)(21,35,27,36)(29,38,34,39)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=g,e^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,d^b=dg,e^b=efg,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #132822, #132823, #132825, … |
| maximal normal subgroup of | #132822, #132823, #132825, … |
| maximal quotient of | #124517, #124533, #124598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4795);