Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 | #2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
774 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)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,22)(21,28)(24,29)(25,26)(27,32)(30,31);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(18,20)(19,25)(21,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,14,15,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32);(1,3)(2,6)(4,7)(5,8)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|b^2=g,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#53210, #8105780, #8105794, … |
| maximal normal subgroup of | C2×#53210, #8105780, #8105794, … |
| maximal quotient of | #606731, #606788, #606795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46198);