Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(10,14)(11,17)(16,21)(18,23)(25,26,29,32)(27,33,35,30)(28,36,38,31)(34,39,37,40);(10,14)(11,17)(12,19)(22,24)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,40)(37,38);(1,3,5,4)(2,7,6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,38)(37,40);(1,4,5,3)(2,8,6,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=gh,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97888, #172715, #172716, … |
| maximal normal subgroup of | #97888, #172715, #172716, … |
| maximal quotient of | #557190, #557226, #557531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38929);