Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C2 | C2×C43 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
735 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(1,2,3,4)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(5,6,7,8)(10,15)(14,20)(16,22)(21,24);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,e^2=h,c^a=ch,d^b=dh136 conjugacy classes — view the full character table.
| Frattini subgroup of | #72257880, #72265296, #72265555 |
| maximal subgroup of | #108432, #108556, #109683, … |
| maximal normal subgroup of | #108432, #108556, #109683, … |
| maximal quotient of | #180786, #180788, #180790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13319);