Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2433 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
787 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 787 conjugacy classes of subgroups — too many to draw. Only its 137 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) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(14,16);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,14)(13,15);(11,15)(13,16)Polycyclic presentation:
2^8|c^2=e,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,f^c=fh,g^c=gh,f^d=fh49 conjugacy classes — view the full character table.
| maximal subgroup of | #248695, #248696, #248701, … |
| maximal normal subgroup of | #248695, #248696, #248701, … |
| maximal quotient of | #349798, #349799, #349803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25978);