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 | |
697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,17)(12,18)(13,16)(14,15)(19,23)(20,24)(21,22)(25,26,30,29)(27,33,35,31)(28,36,38,32)(34,39,37,40);(1,2,5,4)(3,6,8,7)(10,15)(11,18)(16,22)(19,24)(26,29)(31,33)(32,36)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,27)(26,31)(28,37)(29,33)(30,35)(32,40)(34,38)(36,39);(1,2,5,4)(3,6,8,7)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,40)(35,37);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,15,22)(11,19,18,24)(12,20,17,23)(25,29,30,26)(27,33,35,31)(28,36,38,32)(34,40,37,39)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=fg,d^2=fg,e^2=g,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×#41921, #7639866, #7640250, … |
| maximal normal subgroup of | C2×#41921, #7639866, #7640250, … |
| maximal quotient of | #544944, #545329, #545958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34909);