Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(18,22)(23,30)(24,31)(25,32);(2,5)(6,8)(17,18,21,24,20,22,28,31)(19,25,27,30,26,32,29,23);(3,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,26)(21,29)(22,30)(24,32)(25,31)(27,28);(1,3,4,7)(2,6,5,8)(19,26)(23,30)(25,32)(27,29)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=eh,g^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #170256, #170258, #170288, … |
| maximal normal subgroup of | #170256, #170258, #170288, … |
| maximal quotient of | #380815, #380849, #381006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23294);