Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2847 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,20,22,24);(1,2,5,6)(3,4,7,8)(10,13)(14,15)(17,19)(18,20)(21,22)(23,24);(2,6)(3,7)(9,10)(11,15)(12,13)(14,16)(17,20)(18,22)(19,23)(21,24);(1,3,5,7)(2,4,6,8)(9,11,12,16)(10,14,13,15)(17,19)(18,20)(21,22)(23,24);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^8|a^2=fh,b^2=g,d^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #223011, #223039, #223049, … |
| maximal normal subgroup of | #223011, #223039, #223049, … |
| maximal quotient of | #519027, #519084, #519099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29156);