Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 393216 |
| 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 | |
1661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(6,7)(9,10,14,12)(11,16,13,15);(5,6,8,7)(9,11,14,13)(10,15,12,16);(1,2)(3,4)(5,7,8,6)(9,10,14,12)(11,16,13,15);(9,12,14,10)(11,16,13,15);(9,13,14,11)(10,15,12,16)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=h,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #36029⋊C3 |
| maximal subgroup of | #129784, #214789, #214837, … |
| maximal normal subgroup of | #129784, #214789, #214837, … |
| maximal quotient of | #517709, #517743, #517754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29017);