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 | |
738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(10,13)(14,15)(17,18,22,25)(19,23,26,21)(20,24,30,32)(27,31,28,29);(2,5)(3,8)(9,10,12,13)(11,15,16,14)(19,26)(21,23)(27,28)(29,31);(2,5)(6,7)(9,11,12,16)(10,14,13,15)(17,19)(18,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,21,22,23)(18,19,25,26)(20,29,30,31)(24,27,32,28)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^2=fg,e^2=fgh,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×#58913, #8105321, #8105342, … |
| maximal normal subgroup of | C2×#58913, #8105321, #8105342, … |
| maximal quotient of | #585434, #585797, #585938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51901);