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 | 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:
(2,5)(6,8)(10,14)(12,19)(15,20)(18,23)(25,26,30,31)(27,28,35,32)(29,37,38,33)(34,39,40,36);(10,14)(11,17)(16,21)(18,23)(26,31)(28,32)(33,37)(36,39);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(1,3,4,7)(2,6,5,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^8|a^2=f,c^2=fgh,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×#37088, #7634317, #7634388, … |
| maximal normal subgroup of | C2×#37088, #7634317, #7634388, … |
| maximal quotient of | #553955, #553973, #554154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30076);