Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| 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 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #397 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,16,11,14,9,15)(7,12,20,28,18,29,19,27)(8,21,25,31,23,13,24,32)(17,26,34,35,33,36,22,30);(1,2)(3,4)(5,7)(6,12)(8,22,23,34)(9,19)(10,20)(11,18)(13,30,21,35)(14,29)(15,27)(16,28)(17,24,33,25)(26,31,36,32);(2,4)(5,8)(6,13)(7,17)(9,25)(10,24)(11,23)(12,26)(14,21)(15,32)(16,31)(18,33)(19,34)(20,22)(27,35)(28,30)(29,36)Polycyclic presentation:
2^8|a^2=g,b^2=d,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cf,e^b=eh,d^c=dh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11086, #357758, #358688, … |
| maximal normal subgroup of | C2×#11086, #357758, #358688, … |
| maximal quotient of | #115975, #115990, #115991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4074);