Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
765 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)(3,7)(4,8)(5,6)(11,15)(14,16)(17,18,22,23)(19,24,26,21)(20,28,30,25)(27,29,32,31);(1,3,5,8)(2,4,6,7)(9,10,13,12)(11,14,15,16)(18,23)(19,26)(25,28)(27,32);(2,6)(4,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32);(1,4,5,7)(2,3,6,8)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,3)(2,7)(4,6)(5,8)(9,12,13,10)(11,16,15,14)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41661, #7640071, #7640239, … |
| maximal normal subgroup of | C2×#41661, #7640071, #7640239, … |
| maximal quotient of | #544794, #545249, #545858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34649);