Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | #2661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
774 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)(4,5)(6,8)(10,13)(14,16)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31);(1,3,6,4)(2,7,8,5)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32);(2,8)(3,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,4,6,3)(2,5,8,7)(9,11,12,15)(10,14,13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,5)(2,3)(4,8)(6,7)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37569, #7633817, #7633830, … |
| maximal normal subgroup of | C2×#37569, #7633817, #7633830, … |
| maximal quotient of | #556455, #556458, #556460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30557);