Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
792 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)(3,5)(4,7)(6,8)(11,15)(14,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,27)(28,31)(30,32);(9,10)(11,14)(12,13)(15,16)(18,19)(20,27)(24,26)(28,32);(2,3)(6,7)(10,13)(14,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)Polycyclic presentation:
2^8|c^2=f,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #7221586, #7221590, #7221650, … |
| maximal normal subgroup of | #7221586, #7221590, #7221650, … |
| maximal quotient of | #582292, #582494, #582533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45987);