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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(12,20)(15,22)(18,24)(21,23)(26,31)(27,33)(29,37)(36,39);(3,8)(5,7)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26)(27,28)(29,34)(30,31)(32,33)(35,36)(37,38)(39,40);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,29,30,37)(26,34,31,38)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^8|a^2=fg,b^2=gh,c^2=fg,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 | #132756, #487492, #488242, … |
| maximal normal subgroup of | #132756, #487492, #488242, … |
| maximal quotient of | #582244, #582461, #582666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45751);