Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) = C23×#72236189 (order 8192) · Inn(G) = #394 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(6,11)(7,15)(8,9)(12,19)(13,14)(17,18);(5,6,8,11,10,14,9,13)(7,16,17,19,18,20,15,12);(5,7,8,17,10,18,9,15)(6,12,11,16,14,19,13,20)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11253, #358111, C4×#2048, … |
| maximal normal subgroup of | C2×#11253, #358111, C4×#2048, … |
| maximal quotient of | #120178, #120205, #120208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4241);