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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
707 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26,30,33)(27,28,35,31)(29,37,39,32)(34,40,38,36);(4,8)(6,7)(10,11)(12,19)(16,18)(20,24)(25,27,30,35)(26,31,33,28)(29,38,39,34)(32,36,37,40);(2,3)(6,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,3,5,2)(4,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,40)(36,37)(38,39);(1,4,5,8)(2,6,3,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,32)(27,34)(28,36)(30,39)(31,40)(33,37)(35,38)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=h,d^2=fg,e^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×#57863, #8101368, #8101957, … |
| maximal normal subgroup of | C2×#57863, #8101368, #8101957, … |
| maximal quotient of | #586598, #586721, #587086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50851);