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 | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 149 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,21,24,20,22,27,29)(19,23,26,30,25,28,31,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(18,22)(19,25)(24,29)(26,31);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,19,20,25)(18,23,22,28)(21,26,27,31)(24,30,29,32);(2,6)(5,8)(10,13)(14,16)(18,22)(19,25)(24,29)(26,31)Polycyclic presentation:
2^8|a^2=g,c^2=e,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=deh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #136279, #137272, #139977, … |
| maximal normal subgroup of | #136279, #137272, #139977, … |
| maximal quotient of | #357291, #357326, #357333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16305);