Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C23 | C2×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(5,6,7,10,8,11,12,9)(13,14)(15,18)(16,20)(17,19)(21,25)(22,24)(23,26)(27,28);(6,9)(7,12)(10,11)(13,15,16,21,17,22,23,27)(14,18,20,25,19,24,26,28);(1,3,4,2)(14,19)(18,24)(20,26)(25,28)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=deg,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ch,d^b=dg,e^b=eg76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8278, #183369, #183372, … |
| maximal normal subgroup of | C2×#8278, #183369, #183372, … |
| maximal quotient of | #96903, #96904, #105275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1266);