Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 183 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,18,15,12,14,17)(11,19,21,16,23,24,22,20);(2,4)(9,11,14,22,15,23,13,21)(10,16,18,24,12,20,17,19);(6,8)(9,12)(10,15)(11,20)(13,18)(14,17)(16,23)(19,22)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129791, #129792, #137982, … |
| maximal normal subgroup of | #129791, #129792, #137982, … |
| maximal quotient of | #105634, #105635, #105636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3762);