Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1347 conjugacy classes of subgroups; only the 991 normal subgroups are shown.
This group has 1347 conjugacy classes of subgroups — too many to draw. Only its 991 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) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(6,10)(8,15)(11,16)(12,17)(14,19)(18,20);(5,6,8,10,9,12,15,17)(7,11,14,16,13,18,19,20);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20);(7,13)(11,18)(14,19)(16,20);(6,12)(10,17)(11,18)(16,20)Polycyclic presentation:
2^8|a^2=g,b^2=f,f^2=h,b^a=bf,f^a=fh,e^b=eh,d^c=dh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #60109⋊C3 |
| maximal subgroup of | #368074, #368075, #368083, … |
| maximal normal subgroup of | #368074, #368075, #368083, … |
| maximal quotient of | #6344310, #6344323, #6344339, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53097);