Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
729 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 729 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) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,8,12,9,11,17,19)(7,13,15,20,14,10,16,18);(2,4)(5,7)(6,10)(8,16)(9,14)(11,13)(12,20)(15,17)(18,19);(10,13)(18,20)Polycyclic presentation:
2^8|a^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133398, #133399, #133400, … |
| maximal normal subgroup of | #133398, #133399, #133400, … |
| maximal quotient of | #111461, #111557, #111558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3981);