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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(12,19)(16,21)(18,23)(22,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(10,14)(11,17)(16,21)(18,23)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,2,4,5)(3,6,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=gh,d^2=f,e^2=g,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×#37086, #7634338, #7634372, … |
| maximal normal subgroup of | C2×#37086, #7634338, #7634372, … |
| maximal quotient of | #553953, #553974, #554154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30074);