Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | #432 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,20,13,19,29,36,15,21,31,17,30,18,12,27)(11,22,25,37,24,16,23,32,26,33,38,40,39,34,28,35);(1,3,4,7)(2,5,6,8)(9,11,15,26)(10,16)(12,28,29,23)(13,24,30,39)(14,25,31,38)(17,35)(18,22)(19,33)(20,32)(21,34)(27,37)(36,40);(2,6)(5,8)(9,12,13,14,15,29,30,31)(10,17,18,36,21,20,19,27)(11,23,24,38,26,28,39,25)(16,32,22,37,34,35,33,40)Polycyclic presentation:
2^8|a^2=f,b^2=de,c^2=d,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=ch,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93355, #124370, #124371, … |
| maximal normal subgroup of | #93355, #124370, #124371, … |
| maximal quotient of | #135704, #135706, #135707, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5885);