Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | Aut(G) order 32768 |
| 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 | C22×C4 | C24 |
| prime radical | C2×#540 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
569 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 569 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,17,14,16,21,11)(12,15,20,23,19,22,24,18);(9,11,13,16,14,17,21,10)(12,18,20,22,19,23,24,15);(2,5)(6,8)(9,12,14,19)(10,15,16,22)(11,18,17,23)(13,20,21,24);(1,3,4,7)(2,6,5,8)(12,19)(15,22)(18,23)(20,24)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=h,d^2=f,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #215090, #215094, #215098, … |
| maximal normal subgroup of | #215090, #215094, #215098, … |
| maximal quotient of | #207674, #207676, #207692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14416);