Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
674 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)(12,19)(16,21)(18,23)(22,24)(25,26,30,34)(27,32,28,31)(29,37,38,33)(35,39,36,40);(10,14)(11,17)(16,21)(18,23)(25,27,30,28)(26,31,34,32)(29,36,38,35)(33,40,37,39);(1,2,4,5)(3,6,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=gh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37094, #7634359, #7634407, … |
| maximal normal subgroup of | C2×#37094, #7634359, #7634407, … |
| maximal quotient of | #553902, #553917, #554117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30082);