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, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(11,15)(14,16)(17,18,22,23)(19,25,27,21)(20,28,30,24)(26,29,32,31);(2,5)(3,7)(9,10,13,12)(11,14,15,16)(18,23)(21,25)(24,28)(29,31);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,31)(30,32);(9,10,13,12)(11,14,15,16)(17,20,22,30)(18,24,23,28)(19,26,27,32)(21,29,25,31);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,21)(18,19)(20,29)(22,25)(23,27)(24,26)(28,32)(30,31)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=h,d^2=g,e^2=fg,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×#38343, #7635560, #7635667, … |
| maximal normal subgroup of | C2×#38343, #7635560, #7635667, … |
| maximal quotient of | #521742, #522350, #522846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31331);