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 | #2635 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
778 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)(3,5)(4,6)(7,8)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,31,30,32);(4,8)(6,7)(9,10,13,14)(11,12,15,16)(18,22)(20,29)(23,25)(26,30);(2,3)(6,7)(10,14)(11,15)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,15)(14,16)(17,20,21,29)(18,24,22,28)(19,26,27,30)(23,31,25,32);(1,4,5,8)(2,6,3,7)(9,12)(10,11)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=g,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×#55431, #8102263, #8102623, … |
| maximal normal subgroup of | C2×#55431, #8102263, #8102623, … |
| maximal quotient of | #586315, #586323, #586361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48419);