Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(12,20)(16,21)(18,24)(22,23)(25,26)(27,34)(28,32)(29,31)(30,37)(33,38)(35,39)(36,40);(10,14)(15,17)(16,21)(22,23)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,4,5,3)(2,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,d^2=fgh,e^2=g,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.
| Fitting subgroup of | #37613⋊C3 |
| maximal subgroup of | #97051, #241627, #241630, … |
| maximal normal subgroup of | #97051, #241627, #241630, … |
| maximal quotient of | #556523, #556749, #556922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30601);