Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | Aut(G) = C23×#72236189 (order 8192) · Inn(G) = #386 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
415 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 3 generators on 20 points:
(5,6,9,15,11,16,10,14)(7,13,17,20,18,19,8,12);(1,2)(3,4)(5,7)(6,12)(8,10)(9,17)(11,18)(13,15)(14,19)(16,20);(2,4)(5,8,9,18,11,17,10,7)(6,13,14,20,16,19,15,12)Polycyclic presentation:
2^8|a^2=g,c^2=d,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11185, #357941, #358178, … |
| maximal normal subgroup of | C2×#11185, #357941, #358178, … |
| maximal quotient of | #118686, #118687, #118773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4173);