Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,20)(15,18,23,27)(16,19,24,28)(21,25,22,26);(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,21)(17,23)(19,25)(20,27)(22,24)(26,28);(2,4)(5,8,9,12)(6,10,7,11)(15,21)(16,24)(18,25)(19,28)(22,23)(26,27);(2,4)(6,7)(10,11)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=e,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6632314, #6632324, #6632344, … |
| maximal normal subgroup of | #6632314, #6632324, #6632344, … |
| maximal quotient of | #337783, #337784, #337787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23697);