Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10524 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #124 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10,15,19)(11,20,24,35)(12,25,28,17)(13,30,14,18)(16,23,32,22)(21,36,38,33)(26,31,37,40)(27,34,29,39);(1,2)(3,7)(4,6)(5,8)(9,11,15,24)(10,16,19,32)(12,26,28,37)(13,23,14,22)(17,33,25,36)(18,35,30,20)(21,27,38,29)(31,39,40,34);(2,6)(3,4)(7,8)(9,12,13,27,15,28,14,29)(10,17,18,34,19,25,30,39)(11,21,22,37,24,38,23,26)(16,31,20,33,32,40,35,36)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=d,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,f^b=fgh,e^c=egh,f^c=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151281, #151317, C2×#10528, … |
| maximal normal subgroup of | #151281, #151317, C2×#10528, … |
| maximal quotient of | #104881, #104888, #104895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3516);