Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
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 | |
308 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,14)(13,15)(17,18,21,24)(19,22,26,30)(20,23,27,31)(25,29,28,32);(1,3)(2,5)(4,7)(6,8)(9,11,13,12)(10,14,15,16)(19,25)(20,27)(22,29)(23,31)(26,28)(30,32);(2,6)(5,8)(10,15)(11,12)(17,19,21,26)(18,22,24,30)(20,28,27,25)(23,32,31,29);(3,7)(5,8)(9,12,13,11)(10,14,15,16)Polycyclic presentation:
2^8|a^2=efh,b^2=e,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #169488, #169490, #169520, … |
| maximal normal subgroup of | #169488, #169490, #169520, … |
| maximal quotient of | #391287, #391290, #391324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25812);