Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
617 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 617 conjugacy classes of subgroups — too many to draw. Only its 187 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 — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,19,21,24)(12,17)(13,18)(14,16)(15,20,23,22);(2,5)(3,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(10,16)(15,23)(17,18)(19,24);(2,5)(6,8)(9,12,14,13)(10,17,16,18)(11,20,21,22)(15,19,23,24)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=h,f^2=g,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #106184, #107504, #107576, … |
| maximal normal subgroup of | #106184, #107504, #107576, … |
| maximal quotient of | #160581, #160602, #160625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10663);