Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | #2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
728 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)(5,6)(7,8)(9,10,13,12,14,16,22,21)(11,15,19,20,17,23,24,18);(2,4)(9,11)(10,15)(12,20)(13,19)(14,17)(16,23)(18,21)(22,24);(6,8)(11,17)(15,23)(18,20)(19,24);(9,12,13,16,14,21,22,10)(11,18,19,15,17,20,24,23)Polycyclic presentation:
2^8|a^2=g,d^2=g,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #209548, #209550, #265327, … |
| maximal normal subgroup of | #209548, #209550, #265327, … |
| maximal quotient of | #208728, #208729, #208730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14433);