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 | |
691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,17)(12,14)(13,20)(15,18)(16,21)(19,22)(23,24)(25,26,31,29)(27,34,28,32)(30,37,38,33)(35,39,36,40);(3,7)(6,8)(10,12)(11,18)(16,20)(19,24)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=fg,d^2=f,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×#55649, #8100853, #8101173, … |
| maximal normal subgroup of | C2×#55649, #8100853, #8101173, … |
| maximal quotient of | #585737, #586235, #586646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48637);