Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16)(7,17,18,28)(8,22,24,30)(9,15,10,14)(12,21,26,20)(13,25,29,23)(19,33,32,27)(31,35,36,34);(5,7,9,20,11,18,10,21)(6,12,14,17,16,26,15,28)(8,19,23,34,24,32,25,35)(13,27,22,31,29,33,30,36);(1,2,3,4)(7,18)(12,26)(17,28)(19,32)(20,21)(27,33)(31,36)(34,35);(5,8)(6,13)(7,19)(9,23)(10,25)(11,24)(12,27)(14,22)(15,30)(16,29)(17,31)(18,32)(20,34)(21,35)(26,33)(28,36)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,c^b=ch,g^d=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #273693, #273694, #273701, … |
| maximal normal subgroup of | #273693, #273694, #273701, … |
| maximal quotient of | #351772, #351773, #351774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26182);