Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,7,8,6)(10,13)(11,15)(17,18,22,20)(19,23,21,24);(2,5)(3,8)(9,10,12,13)(11,14,15,16)(18,20)(19,21);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,21)(18,23,20,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,23,21,24);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,21,22,19)(18,23,20,24)Polycyclic presentation:
2^8|a^2=gh,b^2=fgh,c^2=fg,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#56289, #8102688, #8103189, … |
| maximal normal subgroup of | C2×#56289, #8102688, #8103189, … |
| maximal quotient of | #585417, #585615, #586575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49277);