Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
279 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 279 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(2,5,6,7)(9,10)(11,14)(12,19,20,15)(13,16)(17,21,23,24)(18,22);(1,2,3,7,4,6,8,5)(9,11,13,18)(10,14,16,22)(12,17,20,23)(15,21,19,24);(9,12,13,20)(10,15,16,19)(11,17,18,23)(14,21,22,24)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=h,d^2=g,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131141, #131142, #131143, … |
| maximal normal subgroup of | #131141, #131142, #131143, … |
| maximal quotient of | #99557, #99558, #99561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2000);