Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | |
689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(3,5)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20)(25,26,31,33)(27,34,28,29)(30,38,39,32)(35,37,36,40);(1,2)(3,7)(4,6)(5,8)(11,19)(13,23)(15,24)(17,20)(25,27,31,28)(26,29,33,34)(30,35,39,36)(32,37,38,40);(1,3,6,5)(2,7,4,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,28,31,27)(26,29,33,34)(30,36,39,35)(32,37,38,40);(1,4,6,2)(3,7,5,8)(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,29)(26,27)(28,33)(30,37)(31,34)(32,35)(36,38)(39,40);(1,5,6,3)(2,7,4,8)(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)(25,30,31,39)(26,32,33,38)(27,35,28,36)(29,37,34,40)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#54226, #8103768, #8103890, … |
| maximal normal subgroup of | C2×#54226, #8103768, #8103890, … |
| maximal quotient of | #585596, #586220, #586758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47214);