Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(6,9)(10,12)(13,15,17,22)(14,18,20,26)(16,23,24,21)(19,27,28,25);(2,4)(13,16,17,24)(14,19,20,28)(15,21,22,23)(18,25,26,27);(14,20)(18,26)(19,28)(25,27);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|b^2=h,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #111662, #112887, #112909, … |
| maximal normal subgroup of | #111662, #112887, #112909, … |
| maximal quotient of | #515047, #515100, #515108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27590);