Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
147 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)(7,17,21,35)(8,23)(9,13)(10,28)(11,16)(12,19,30,18)(14,25)(15,24)(20,33,22,34)(26,31,36,29)(27,32);(5,7,10,20,11,21,24,22)(6,12,15,29,16,30,28,31)(8,18,9,26,25,19,27,36)(13,17,23,34,32,35,14,33);(1,2,3,4)(5,8,24,27,11,25,10,9)(6,13,28,14,16,32,15,23)(7,18,22,36,21,19,20,26)(12,17,31,33,30,35,29,34)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=eh,d^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136420, #136422, #136509, … |
| maximal normal subgroup of | #136420, #136422, #136509, … |
| maximal quotient of | #140391, #140406, #140422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6175);