Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,13,17,14,12,16,21)(11,15,20,23,18,19,22,24);(2,4)(5,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,22)(17,23)(21,24);(5,7,8,6)(10,12)(11,18)(17,21)(20,22);(9,12,13,21,14,10,16,17)(11,19,20,24,18,15,22,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=f,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 | #493228, #493320, #494953, … |
| maximal normal subgroup of | #493228, #493320, #494953, … |
| maximal quotient of | #183851, #183863, #184053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13481);