Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14,18,21,17,20,26,16)(15,22,25,24,23,19,27,28);(1,2,4,3)(6,10)(11,12)(13,15,17,23)(14,19,20,22)(16,24,21,28)(18,25,26,27);(1,3,4,2)(5,7,9,8)(6,11,10,12)(14,20)(15,23)(16,21)(25,27);(5,8,9,7)(6,12,10,11)(13,16,26,20,17,21,18,14)(15,24,27,22,23,28,25,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=eg,d^2=ef,f^2=h,b^a=be,c^a=ch,c^b=ch,d^c=dh100 conjugacy classes — view the full character table.
| maximal subgroup of | #128576, #128578, #128594, … |
| maximal normal subgroup of | #128576, #128578, #128594, … |
| maximal quotient of | #184373, #184381, #184387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13509);