Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 4096 |
| 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 | C4×D8 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
215 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 215 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,16,12,17,11,27)(7,18,21,31,24,36,23,29)(8,14,25,33,26,34,10,15)(13,28,30,19,32,22,35,20);(2,4)(5,7,10,22,11,23,26,19,12,24,25,28,9,21,8,20)(6,13,14,31,16,30,33,36,17,32,34,29,27,35,15,18);(5,8,9,25,12,26,11,10)(6,14,16,33,17,34,27,15)(7,19,21,22,24,20,23,28)(13,29,30,18,32,31,35,36)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=gh,e^2=h,f^2=g,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 | #128694, #128713, #135910, … |
| maximal normal subgroup of | #128694, #128713, #135910, … |
| maximal quotient of | #128534, #128589, #128602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5630);