Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10)(11,15)(12,20,22,16)(13,17)(14,18)(19,24,21,23);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20);(2,5)(6,8)(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=e,d^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #139289, #139291, #139297, … |
| maximal normal subgroup of | #139289, #139291, #139297, … |
| maximal quotient of | #222884, #222885, #222891, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15466);