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 | #2543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,5)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,24,26,21)(20,28,29,25)(27,30,32,31);(1,2,6,7)(3,4,5,8)(11,16)(13,15)(18,23)(19,26)(20,29)(30,31);(1,3,6,5)(2,8,7,4)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32);(1,4,6,8)(2,3,7,5)(9,10)(11,13)(12,14)(15,16)(17,20,22,29)(18,25,23,28)(19,27,26,32)(21,30,24,31);(1,5,6,3)(2,8,7,4)(17,21,22,24)(18,19,23,26)(20,30,29,31)(25,27,28,32)Polycyclic presentation:
2^8|a^2=fg,b^2=fgh,c^2=g,d^2=f,e^2=f,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 | #132610, #487572, #488402, … |
| maximal normal subgroup of | #132610, #487572, #488402, … |
| maximal quotient of | #582254, #582460, #582665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45749);