Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C23×D8 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
749 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 749 conjugacy classes of subgroups — too many to draw. Only its 135 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,13,18,14,16,12,17)(11,19,20,23,22,15,21,24);(2,4)(5,6)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23);(6,8)(10,16)(11,20)(12,13)(15,23)(19,24)(21,22)Polycyclic presentation:
2^8|a^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93898, #137374, #137375, … |
| maximal normal subgroup of | #93898, #137374, #137375, … |
| maximal quotient of | #111221, #111275, #111279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4015);