Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26)(27,32)(28,29)(30,31)(33,38)(34,40)(35,37)(36,39);(1,2,4,6)(3,5,7,8)(10,15)(11,17)(16,22)(18,23)(26,31)(29,33)(32,39)(35,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,40)(34,38);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,28)(26,33)(27,34)(29,31)(30,38)(32,35)(36,37)(39,40);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,29,30,33)(26,28,31,38)(27,35,36,40)(32,37,39,34)Polycyclic presentation:
2^8|a^2=h,b^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41396, #7639909, #7640012, … |
| maximal normal subgroup of | C2×#41396, #7639909, #7640012, … |
| maximal quotient of | #544455, #545041, #545527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34384);