Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,24,21,26,20,25)(19,23,27,32,29,30,28,31);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19,20,28,21,29,22,27)(18,23,25,31,26,30,24,32);(10,14)(13,16)(18,24)(19,27)(20,22)(23,30)(25,26)(28,29);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=gh,b^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #311004, #311012, #311020, … |
| maximal normal subgroup of | #311004, #311012, #311020, … |
| maximal quotient of | #360396, #362552, #366060, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20716);