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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
741 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,12)(13,16)(14,15)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,28,31,32);(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,14,15,12)(20,27)(23,30)(25,31)(28,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,16,12)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,26)(18,22,24,29)(20,25,27,31)(23,28,30,32);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,15,16,11)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=fh,d^2=fg,e^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×#53339, #8105825, #8105878, … |
| maximal normal subgroup of | C2×#53339, #8105825, #8105878, … |
| maximal quotient of | #606807, #606825, #606845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46327);