Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1405 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 1405 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,9)(4,7)(5,8)(6,10)(11,15)(12,14)(13,16);(1,3,4,10)(2,6,7,9)(5,11,12,16)(8,13,14,15);(6,13)(9,15);(2,7)(6,9)(8,14)(13,15)Polycyclic presentation:
2^8|b^2=e,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139622, #139624, #139644, … |
| maximal normal subgroup of | #139622, #139624, #139644, … |
| maximal quotient of | #251159, #251160, #251161, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16972);