Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 91 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:
(2,4)(5,6,7,8)(9,10)(11,19)(12,22)(13,16)(14,24)(15,18)(17,20)(21,23);(6,8)(9,11,20,16,14,21,23,24,13,17,19,10,15,22,12,18);(1,2,3,4)(10,16)(11,17)(18,24)(21,22)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eh,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #131857, #131887, #136883, … |
| maximal normal subgroup of | #131857, #131887, #136883, … |
| maximal quotient of | #140523, #140572, #140901, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6194);