Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C2×#120 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,10,11)(7,9,8,12)(14,18)(15,21)(20,25)(22,27);(6,11)(7,8)(14,18)(19,24)(20,25)(26,28);(5,7,10,8)(6,12,11,9)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(1,3,4,2)(5,8,10,7)(6,9,11,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(5,9)(6,7)(8,11)(10,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#34504, #7626106, #7626128, … |
| maximal normal subgroup of | C2×#34504, #7626106, #7626128, … |
| maximal quotient of | #513837, #513856, #513864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27492);