Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1013 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,15)(8,14)(9,12)(10,13)(11,16)(17,19)(18,20);(2,4)(5,7)(6,11)(8,17)(9,16)(10,18)(12,20)(13,19)(14,15);(6,12)(7,16)(8,9)(13,14)(15,20)(17,18);(6,13)(11,19)(12,14)(15,20)Polycyclic presentation:
2^8|e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,g^b=gh,e^c=eh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #376201, #376203, #381483, … |
| maximal normal subgroup of | #376201, #376203, #381483, … |
| maximal quotient of | #232445, #232447, #232449, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16427);