Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,8)(7,11,12,10);(6,8)(10,11)(13,14)(15,16)(17,18)(19,20);(1,2)(3,4)(7,12)(10,11)(14,18)(15,19);(5,7)(6,10)(8,11)(9,12)(13,15,17,19)(14,16,18,20);(5,8,9,6)(7,11,12,10)(13,16)(14,19)(15,18)(17,20)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41941, #7640448, #7640459, … |
| maximal normal subgroup of | C2×#41941, #7640448, #7640459, … |
| maximal quotient of | #544482, #545058, #545412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34929);