Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) order 65536 |
| 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 | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 107 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:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,31)(29,30);(2,5)(6,8)(19,26)(21,28)(22,30)(24,31)(25,32)(27,29);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,19,20,27)(18,22,23,25)(21,29,28,26)(24,32,31,30);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(18,23)(22,25)(24,31)(30,32)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=fh,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #150356, #150698, C2×#30049, … |
| maximal normal subgroup of | #150356, #150698, C2×#30049, … |
| maximal quotient of | #331062, #331063, #331064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23037);