Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,14)(15,16)(17,18,23,26)(19,27,22,24)(20,30,25,21)(28,32,31,29);(2,5)(3,7)(17,19,23,22)(18,24,26,27)(20,31,25,28)(21,29,30,32);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,20,23,25)(18,21,26,30)(19,28,22,31)(24,32,27,29);(9,11,13,12)(10,15,14,16)(17,21)(18,25)(19,29)(20,26)(22,32)(23,30)(24,28)(27,31);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,16,14,15)(17,22,23,19)(18,24,26,27)(20,31,25,28)(21,32,30,29)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#38303, #7635593, #7635667, … |
| maximal normal subgroup of | C2×#38303, #7635593, #7635667, … |
| maximal quotient of | #521732, #522342, #522852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31291);