Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
221 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 95 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:
(1,2,4,3)(5,6)(7,16)(8,21)(9,24)(10,26)(11,15)(12,18)(13,23)(14,22)(17,30)(19,27)(20,36)(25,29)(28,35)(31,34)(32,33);(5,7,9,19,22,33,23,34,11,17,8,18,10,20,25,35)(6,12,13,16,26,31,24,36,15,27,29,30,14,28,21,32);(1,3,4,2)(7,17)(12,27)(16,30)(18,19)(20,33)(28,31)(32,36)(34,35)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=gh,f^2=gh,g^2=h,b^a=bd,d^a=dg,f^a=fgh,g^a=gh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #128690, #128721, #128740, … |
| maximal normal subgroup of | #128690, #128721, #128740, … |
| maximal quotient of | #128531, #128532, #128587, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5628);