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 | |
793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(5,8)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32);(1,2,4,6)(3,5,7,8)(11,16)(13,15)(18,21)(19,25)(20,28)(31,32);(1,3,4,7)(2,5,6,8)(9,11,12,16)(10,13,14,15)(17,19,22,25)(18,23,21,27)(20,29,28,26)(24,32,30,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,28)(18,24,21,30)(19,26,25,29)(23,31,27,32);(1,2)(3,5)(4,6)(7,8)(17,21,22,18)(19,27,25,23)(20,30,28,24)(26,32,29,31)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=fg,d^2=f,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×#55703, #8099224, #8099238, … |
| maximal normal subgroup of | C2×#55703, #8099224, #8099238, … |
| maximal quotient of | #585607, #586475, #586631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48691);