Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) = C23×#72236189 (order 8192) · Inn(G) = #386 · Out(G) = C27 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6)(7,15)(8,13)(9,12)(10,14)(11,16)(17,20)(18,19);(5,7)(6,11,14,19)(8,16)(9,18)(10,17)(12,15,13,20);(1,3,4,2)(6,12)(8,9)(11,15)(13,14)(16,18)(19,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,e^b=eh,d^c=dh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10903, #353936, C4×#1904, … |
| maximal normal subgroup of | C2×#10903, #353936, C4×#1904, … |
| maximal quotient of | #109534, #109535, #109538, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3891);