Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,18)(11,12,16,15)(13,21,22,17)(19,23,24,20)(25,26)(27,34)(28,29)(30,38)(31,33)(32,39)(35,37)(36,40);(11,16)(12,15)(19,24)(20,23)(25,27,31,28)(26,29,33,34)(30,35,39,36)(32,37,38,40);(1,2,4,5)(3,6,7,8)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,28,31,27)(26,29,33,34)(30,36,39,35)(32,37,38,40);(1,3,4,7)(2,6,5,8)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,29)(26,27)(28,33)(30,37)(31,34)(32,35)(36,38)(39,40);(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,30,31,39)(26,32,33,38)(27,35,28,36)(29,37,34,40)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6486329, #6486338, #6486342, … |
| maximal normal subgroup of | #6486329, #6486338, #6486342, … |
| maximal quotient of | #553823, #553837, #554054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30392);