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) = C28 |
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 | |
947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(21,22,26,29)(23,24,31,27)(25,33,34,28)(30,35,36,32);(6,10)(7,13)(8,16)(18,20)(21,23,26,31)(22,27,29,24)(25,30,34,36)(28,35,33,32);(1,2)(3,4)(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18)(21,24)(22,23)(25,32)(26,27)(28,30)(29,31)(33,36)(34,35);(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,23)(22,24)(25,30)(26,31)(27,29)(28,32)(33,35)(34,36);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,25,26,34)(22,28,29,33)(23,30,31,36)(24,32,27,35)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=h,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×#51321, #7706880, #7706884, … |
| maximal normal subgroup of | C2×#51321, #7706880, #7706884, … |
| maximal quotient of | #560359, #560402, #561453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44309);