Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| 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 | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#466 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 175 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,14,13,15,18,12,17)(11,19,20,21,22,23,24,16);(5,7,8,6)(9,11)(10,16)(12,20)(13,23)(14,24)(15,22)(17,19)(18,21);(2,4)(9,12,15,14)(10,17,18,13)(11,20,22,24)(16,19,21,23);(9,13,14,18,15,17,12,10)(11,21,20,23,22,16,24,19)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=h,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #498122, #498233, #498377, … |
| maximal normal subgroup of | #498122, #498233, #498377, … |
| maximal quotient of | #229111, #229129, #229153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16019);