Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
537 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 537 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)(7,13,16,11)(8,17)(9,18)(10,12)(14,15,20,19);(1,2,3,4)(5,7)(6,11)(8,14)(9,19)(10,16)(12,13)(15,17)(18,20);(5,8,17,6,10,18,9,12)(7,14,15,11,16,20,19,13)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=efh,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,e^b=eh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130114, #130116, #130118, … |
| maximal normal subgroup of | #130114, #130116, #130118, … |
| maximal quotient of | #110051, #110053, #110057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3936);