Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,7,4,8)(9,10,12,13)(11,15,16,14)(17,18,21,22)(19,24,20,23);(2,6)(7,8)(10,13)(14,15)(18,22)(19,20);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,19,21,20)(18,23,22,24);(17,20,21,19)(18,23,22,24);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=h,e^2=fg,b^a=bf,c^a=cg,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #224554, #224560, #224566, … |
| maximal normal subgroup of | #224554, #224560, #224566, … |
| maximal quotient of | #514146, #514181, #514348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27906);