Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 163 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 — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,21,16,22,15,20)(11,23,26,17,28,36,27,37)(12,29,13,19,32,38,31,18)(24,34,25,40,30,35,33,39);(2,5)(6,8)(9,11,15,27,16,28,14,26)(10,17,21,36,22,37,20,23)(12,30,31,25,32,24,13,33)(18,35,29,39,19,34,38,40);(1,3,4,7)(2,6,5,8)(9,12,14,13,16,32,15,31)(10,18,20,38,22,19,21,29)(11,24,26,25,28,30,27,33)(17,34,23,39,37,35,36,40);(9,13,14,32,16,31,15,12)(10,19,21,38,22,18,20,29)(11,25,26,30,28,33,27,24)(17,35,36,39,37,34,23,40)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=e,d^2=g,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ch,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124619, #128890, #134568, … |
| maximal normal subgroup of | #124619, #128890, #134568, … |
| maximal quotient of | #212615, #212616, #212617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14651);