Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(11,17)(15,20)(18,23)(22,24)(25,26,29,33)(27,31,35,30)(28,36,37,32)(34,38,40,39);(10,14)(11,17)(12,19)(22,24)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39);(1,4,5,3)(2,8,6,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,37)(38,40)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46309, #7702994, #7703004, … |
| maximal normal subgroup of | C2×#46309, #7702994, #7703004, … |
| maximal quotient of | #557187, #557199, #557514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39297);