Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
441 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 441 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,16,12,8,10,15)(7,17,21,33,22,34,20,31)(9,24,27,23,28,32,26,14)(13,25,30,36,18,35,29,19);(5,7)(6,13)(8,18)(9,25)(10,20)(11,21)(12,22)(14,31)(15,29)(16,30)(17,24)(19,26)(23,33)(27,36)(28,35)(32,34);(2,4)(5,8,12,6)(7,18,22,13)(9,23,28,14)(10,15,11,16)(17,19,34,36)(20,29,21,30)(24,26,32,27)(25,33,35,31)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,g^b=gh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12749, #493550, #493552, … |
| maximal normal subgroup of | C2×#12749, #493550, #493552, … |
| maximal quotient of | #129521, #129563, #129792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5737);