Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,14,18)(11,15,21,24)(12,16,13,17)(19,22,20,23)(25,26)(27,34)(28,32)(29,31)(30,35)(33,37)(36,40)(38,39);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,23)(17,22)(18,24)(25,27,28,35)(26,30,32,34)(29,38,37,36)(31,40,33,39);(1,4,5,3)(2,6,7,8)(10,16)(11,19)(12,13)(17,18)(20,21)(22,23)(26,31)(29,37)(30,39)(32,33)(34,40)(36,38)Polycyclic presentation:
2^8|a^2=dg,b^2=d,c^2=dh,e^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #10586⋊C3 |
| maximal subgroup of | #94729, C2×#10586, #241236, … |
| maximal normal subgroup of | #94729, C2×#10586, #241236, … |
| maximal quotient of | #105175, #105183, #120081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3574);