Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 65536 |
| 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 | |
944 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)(5,6,10,13)(7,11,14,19)(8,9,12,18)(15,20,17,16)(21,22)(23,25)(24,28)(26,34)(27,30)(29,35)(31,36)(32,33);(7,14)(8,12)(9,18)(11,19)(21,23,27,24)(22,28,30,25)(26,31,35,32)(29,36,34,33);(1,2,3,4)(5,7)(6,11)(8,17)(9,16)(10,14)(12,15)(13,19)(18,20)(21,24,27,23)(22,28,30,25)(26,32,35,31)(29,36,34,33);(5,8,10,12)(6,9,13,18)(7,15,14,17)(11,20,19,16)(21,25,27,28)(22,23,30,24)(26,33,35,36)(29,31,34,32);(5,9)(6,12)(7,16)(8,13)(10,18)(11,15)(14,20)(17,19)(21,26,27,35)(22,29,30,34)(23,31,24,32)(25,33,28,36)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,d^2=fh,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 | #132420, #245954, #245955, … |
| maximal normal subgroup of | #132420, #245954, #245955, … |
| maximal quotient of | #560467, #560486, #561564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43886);