Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
265 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 83 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:
(2,4)(5,6)(7,8)(9,10,14,19,29,38,27,36,15,20,13,18,12,17,28,37)(11,21,24,34,22,40,30,16,26,33,23,32,25,39,31,35);(1,2,3,4)(9,11)(10,16)(12,22)(13,30)(14,31)(15,26)(17,32)(18,39)(19,40)(20,35)(21,36)(23,27)(24,28)(25,29)(33,37)(34,38);(6,8)(9,12,15,29)(10,17,20,38)(11,22,26,25)(13,27,14,28)(16,32,35,34)(18,36,19,37)(21,40,33,39)(23,31,24,30)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=h,d^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131333, #131335, #131409, … |
| maximal normal subgroup of | #131333, #131335, #131409, … |
| maximal quotient of | #136606, #136634, #136881, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5982);