Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#120 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
725 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 725 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,8)(9,13)(10,14)(11,15)(12,19)(16,18)(17,20);(1,2)(3,4)(5,7)(6,8)(9,16)(10,12)(11,17)(13,18)(14,19)(15,20);(1,3,4,2)(5,8)(6,12)(7,13)(9,18)(10,19)(11,20)(14,17)(15,16);(5,9,11,10)(6,13,15,14)(7,16,17,12)(8,18,20,19)Polycyclic presentation:
2^8|c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137170, #137171, #137179, … |
| maximal normal subgroup of | #137170, #137171, #137179, … |
| maximal quotient of | #238435, #238438, #238439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16486);