Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,18)(11,15,20,24)(12,17,13,22)(16,21,23,19);(1,2,3,4)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24);(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23);(6,8)(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)Polycyclic presentation:
2^8|a^2=h,b^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137998, #137999, #138054, … |
| maximal normal subgroup of | #137998, #137999, #138054, … |
| maximal quotient of | #347769, #347772, #347775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25904);