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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,8)(7,11,14,15)(9,12,18,16)(13,19,17,20)(21,22,26,31)(23,28,33,27)(24,29,34,35)(25,32,36,30);(5,7,10,14)(6,11,8,15)(9,17,18,13)(12,20,16,19)(21,23,26,33)(22,27,31,28)(24,32,34,30)(25,35,36,29);(1,2,3,4)(5,7)(6,11)(8,15)(9,13)(10,14)(12,19)(16,20)(17,18)(21,23)(22,28)(24,30)(25,29)(26,33)(27,31)(32,34)(35,36);(5,8,10,6)(7,11,14,15)(9,16,18,12)(13,19,17,20)(21,24)(22,29)(23,32)(25,27)(26,34)(28,36)(30,33)(31,35);(5,9)(6,12)(7,13)(8,16)(10,18)(11,19)(14,17)(15,20)(21,25,26,36)(22,30,31,32)(23,29,33,35)(24,27,34,28)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,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 | #132436, #245956, #245982, … |
| maximal normal subgroup of | #132436, #245956, #245982, … |
| maximal quotient of | #560466, #560473, #561561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44194);