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) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(6,8)(9,10)(11,14)(12,13)(15,16);(2,4)(10,13)(14,15);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16);(2,4)(9,11,12,16)(10,14,13,15);(1,2)(3,4)(9,10)(11,15)(12,13)(14,16)Polycyclic presentation:
2^8|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 | #219850, #219864, #219890, … |
| maximal normal subgroup of | #219850, #219864, #219890, … |
| maximal quotient of | #519001, #519013, #519226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29084);