Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
533 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 533 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,13,10,8,12,17)(7,14,16,20,15,11,18,19);(1,2,3,4)(5,7)(6,11)(8,14)(9,18)(10,15)(12,16)(13,20)(17,19);(5,8,9,17,10,6,12,13)(7,14,18,19,15,11,16,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^b=eh,f^b=fh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #130106, #130107, #130108, … |
| maximal normal subgroup of | #130106, #130107, #130108, … |
| maximal quotient of | #110044, #110045, #110046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3925);