Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #390 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,17)(8,22,26,13)(9,16)(10,14)(11,15)(12,19)(18,33,35,30)(20,28)(21,29)(23,36,34,27)(24,31,25,32);(1,2,3,4)(5,7,9,19,11,21,10,20)(6,12,14,17,15,28,16,29)(8,23,24,18,26,34,25,35)(13,30,31,27,22,33,32,36);(5,8)(6,13)(7,18)(9,25)(10,24)(11,26)(12,27)(14,32)(15,22)(16,31)(17,30)(19,23)(20,34)(21,35)(28,36)(29,33)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^c=dh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11126, #357809, #358716, … |
| maximal normal subgroup of | C2×#11126, #357809, #358716, … |
| maximal quotient of | #116985, #116986, #116999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4114);