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 | |
692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,26,31,34)(27,32,28,29)(30,33,38,40)(35,39,36,37);(1,3)(2,6)(4,5)(7,8)(10,15)(12,16)(18,21)(20,23)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(3,4)(6,8)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,28,31,27)(26,32,34,29)(30,36,38,35)(33,39,40,37);(1,4,5,3)(2,6,7,8)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29,31,32)(26,28,34,27)(30,37,38,39)(33,36,40,35);(1,2)(3,8)(4,6)(5,7)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fgh,d^2=fg,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×#56819, #8099569, #8100300, … |
| maximal normal subgroup of | C2×#56819, #8099569, #8100300, … |
| maximal quotient of | #586109, #586248, #587167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49807);