Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 151 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 — 4 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,24,22,20);(2,6)(7,8)(9,11,12,15)(10,13,14,16);(1,3,5,4)(2,7,6,8)(17,19,20,18,21,22,24,23);(1,4,5,3)(2,8,6,7)(10,14)(13,16)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #102174, #102180, #102182, … |
| maximal normal subgroup of | #102174, #102180, #102182, … |
| maximal quotient of | #392531, #393177, #393267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26266);