Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | C2×#121 | C22 |
| 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)(9,10,13,11)(12,15,16,14)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(1,2,4,6)(3,5,7,8)(9,11,13,10)(12,15,16,14)(18,22)(19,26)(20,29)(30,31);(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(1,2)(3,5)(4,6)(7,8)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(1,3,4,7)(2,5,6,8)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#43519, #7667765, #7668038, … |
| maximal normal subgroup of | C2×#43519, #7667765, #7668038, … |
| maximal quotient of | #535375, #535676, #535999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 36507);