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×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
467 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 467 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,15,11,16,10,27)(7,17,19,32,20,31,26,13)(8,21,23,33,25,35,24,14)(12,22,28,34,30,36,29,18);(5,7)(6,12)(8,22)(9,19)(10,26)(11,20)(13,14)(15,28)(16,30)(17,21)(18,24)(23,34)(25,36)(27,29)(31,35)(32,33);(2,4)(5,7)(6,13)(8,18)(9,26)(10,19)(11,20)(12,14)(15,31)(16,32)(17,27)(21,29)(22,24)(23,36)(25,34)(28,35)(30,33)Polycyclic presentation:
2^8|a^2=f,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×#12748, #493550, #493551, … |
| maximal normal subgroup of | C2×#12748, #493550, #493551, … |
| maximal quotient of | #129520, #129564, #129791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5736);