Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
913 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)(3,5)(4,7)(6,8)(9,10,15,13)(11,16,14,12)(17,18,20,22)(19,21,23,24);(2,3)(6,7)(9,11,15,14)(10,16,13,12);(9,12)(10,14)(11,13)(15,16)(17,18)(19,21)(20,22)(23,24);(1,3,5,2)(4,7,8,6)(9,13,15,10)(11,16,14,12);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,16,13,12)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37214, #7634547, #7634557, … |
| maximal normal subgroup of | C2×#37214, #7634547, #7634557, … |
| maximal quotient of | #553923, #553925, #554141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30202);