Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
559 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 559 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,9)(8,16)(10,15)(11,13)(12,19)(14,18)(17,20);(1,3,4,2)(5,7,9,6,11,15,10,13)(8,14,17,12,19,20,18,16);(5,8,10,18,11,19,9,17)(6,12,7,14,13,16,15,20);(5,9,11,10)(6,7,13,15)(8,17,19,18)(12,14,16,20)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=e,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #360290, #367941, #368474, … |
| maximal normal subgroup of | #360290, #367941, #368474, … |
| maximal quotient of | #202481, #202487, #202493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14066);