Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C23 |
| commutator | C23 | C22×C8 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 439 conjugacy classes of subgroups — too many to draw. Only its 179 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,16)(12,14)(13,15)(17,18,19,21,20,22,23,24);(2,4)(5,6)(7,8)(9,11,13,12)(10,14,15,16);(2,4)(6,8)(9,12,13,11)(10,14,15,16)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,g^2=h,b^a=bd,c^a=ce,c^b=cf88 conjugacy classes — view the full character table.
| maximal subgroup of | #92816, #92887, #93497, … |
| maximal normal subgroup of | #92816, #92887, #93497, … |
| maximal quotient of | #121614, #121615, #121616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4334);