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 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(10,13)(11,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32);(2,3)(6,7)(9,10,12,13)(11,16,15,14)(18,21)(19,25)(20,28)(31,32);(9,10,12,13)(11,14,15,16)(17,19,22,25)(18,23,21,27)(20,29,28,26)(24,32,30,31);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,28)(18,24,21,30)(19,26,25,29)(23,31,27,32);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,21,22,18)(19,27,25,23)(20,30,28,24)(26,32,29,31)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=gh,d^2=fg,e^2=f,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×#56830, #8100225, #8100300, … |
| maximal normal subgroup of | C2×#56830, #8100225, #8100300, … |
| maximal quotient of | #585887, #586285, #586669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49818);