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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,26,29,30)(27,33,34,31)(28,32,37,38)(35,40,36,39);(1,2,5,4)(3,6,7,8)(10,14)(15,17)(16,20)(22,23)(26,30)(27,34)(32,38)(35,36);(1,3)(2,6)(4,8)(5,7)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,27,29,34)(26,31,30,33)(28,36,37,35)(32,40,38,39);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,29,37)(26,32,30,38)(27,35,34,36)(31,39,33,40);(1,4,5,2)(3,8,7,6)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,40)(37,38)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=f,d^2=h,e^2=fg,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×#50976, #7706638, #7706691, … |
| maximal normal subgroup of | C2×#50976, #7706638, #7706691, … |
| maximal quotient of | #560858, #561082, #561945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43964);