Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
674 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)(9,10,14,17)(11,13,16,22)(12,15,19,23)(18,21,20,24)(25,26)(27,34)(28,36)(29,32)(30,31)(33,38)(35,39)(37,40);(11,16)(12,19)(13,22)(15,23)(25,27,31,29)(26,32,30,34)(28,37,38,35)(33,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,13,17,22)(12,20,19,18)(15,24,23,21)(25,28,31,38)(26,33,30,36)(27,35,29,37)(32,39,34,40);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,18)(13,21)(14,19)(16,20)(17,23)(22,24)(25,29,31,27)(26,32,30,34)(28,35,38,37)(33,40,36,39);(9,13)(10,16)(11,17)(12,21)(14,22)(15,20)(18,23)(19,24)(25,30,31,26)(27,32,29,34)(28,36,38,33)(35,39,37,40)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#50205, #7706031, #7706214, … |
| maximal normal subgroup of | C2×#50205, #7706031, #7706214, … |
| maximal quotient of | #561296, #561398, #562448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43193);