Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C23 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(6,9)(7,11)(10,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(5,6,7,9,8,10,11,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(1,2,3,4)(6,10)(9,12)(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^8|a^2=f,b^2=d,c^2=e,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=cgh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93740, #99413, #99414, … |
| maximal normal subgroup of | #93740, #99413, #99414, … |
| maximal quotient of | #99310, #99311, #99312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1800);