Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C2×#72235851 (order 2048) · Inn(G) = C2×D4 · Out(G) = C2×#525 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | #525 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 85 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)(5,6,9,14,10,15,24,32,11,16,25,33,8,13,18,27)(7,17,20,34,19,29,35,31,21,30,23,36,22,12,26,28);(5,7,18,35,8,22,25,20,11,21,24,26,10,19,9,23)(6,12,27,34,13,30,33,28,16,29,32,36,15,17,14,31);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^b=dh,f^b=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129573, #129574, #129597, … |
| maximal normal subgroup of | #129573, #129574, #129597, … |
| maximal quotient of | #136573, #136574, #136581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5966);