Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(7,8)(9,10,13,16)(11,14,12,15)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32);(1,2,6,3)(4,7,5,8)(9,10)(11,14)(12,15)(13,16)(19,25)(23,26)(27,31)(30,32);(1,3,6,2)(4,8,5,7)(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,4,6,5)(2,7,3,8)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32);(1,5,6,4)(2,8,3,7)(9,12,13,11)(10,15,16,14)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=f,d^2=gh,e^2=fh,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #132448, #253832, #253834, … |
| maximal normal subgroup of | #132448, #253832, #253834, … |
| maximal quotient of | #517896, #517898, #517909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29040);