Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| 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 | |
165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,15)(8,20)(9,14)(10,12)(11,17)(13,21)(16,24)(18,25)(19,28)(22,34)(23,26)(27,30)(29,32)(31,36)(33,35);(2,4)(5,7,8,18,9,19,22,33,10,17,21,32,23,16,30,31)(6,11,13,25,14,24,27,35,12,15,20,29,26,28,34,36);(6,12)(7,16,17,19)(8,21)(11,24,15,28)(14,26)(18,33,32,31)(22,30)(25,36,29,35)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,f^2=g,g^2=h,b^a=bd,c^a=ch,f^a=fh,c^b=cg,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93614, #93615, #93651, … |
| maximal normal subgroup of | #93614, #93615, #93651, … |
| maximal quotient of | #129056, #129069, #129079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5648);