Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 151 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)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,19,21,24,30,26)(20,23,29,25,28,31,32,27);(1,3)(2,5)(4,8)(6,7)(17,19,22,24,21,26,30,18)(20,27,29,23,28,25,32,31);(3,8)(5,7)(10,13)(14,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,31)(26,27)(30,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,24)(19,26)(20,28)(29,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #140054, #140060, C2×#22320, … |
| maximal normal subgroup of | #140054, #140060, C2×#22320, … |
| maximal quotient of | #355306, #355636, #355685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15308);