Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
497 conjugacy classes of subgroups; only the 343 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 343 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:
(1,2,4,3)(5,6,9,14,11,16,10,15)(7,17,19,26,20,30,24,12)(8,13,22,28,21,27,23,29)(18,31,33,35,32,36,34,25);(5,7,10,19,11,20,9,24)(6,12,15,17,16,26,14,30)(8,18,23,33,21,32,22,34)(13,25,29,31,27,35,28,36);(1,3,4,2)(5,8,11,21)(6,13,16,27)(7,18,20,32)(9,22,10,23)(12,25,26,35)(14,28,15,29)(17,31,30,36)(19,33,24,34);(8,21)(13,27)(18,32)(22,23)(25,35)(28,29)(31,36)(33,34)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=h,g^2=h,b^a=be,e^b=eh,f^b=fh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #135269, #138288, #138289, … |
| maximal normal subgroup of | #135269, #138288, #138289, … |
| maximal quotient of | #183366, #183374, #183381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13447);