Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1438 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,26,28,32)(14,25)(16,24,31,29)(19,22)(21,30,23,27);(1,3)(2,7)(4,12)(5,19)(6,14)(8,23)(9,25)(10,22)(11,18)(13,20)(15,21)(16,30)(17,28)(24,26)(27,31)(29,32);(1,4)(2,8)(3,12)(5,16)(6,17)(7,21)(9,24)(10,15)(11,26)(13,27)(14,28)(18,29)(19,30)(20,31)(22,23)(25,32)Polycyclic presentation:
2^8|a^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11000, #358715, #359534, … |
| maximal normal subgroup of | C2×#11000, #358715, #359534, … |
| maximal quotient of | #111253, #111254, #111255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3988);