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 | |
948 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)(5,6,10,11)(7,8,12,16)(9,17,18,13)(14,20,19,15)(21,22,26,29)(23,27,30,35)(24,25,28,34)(31,36,33,32);(6,11)(7,12)(9,18)(15,20)(23,30)(24,28)(25,34)(27,35);(1,2,3,4)(5,7)(6,8)(9,14)(10,12)(11,16)(13,15)(17,20)(18,19)(21,23)(22,27)(24,33)(25,32)(26,30)(28,31)(29,35)(34,36);(5,8)(6,12)(7,11)(9,15)(10,16)(13,19)(14,17)(18,20)(21,24,26,28)(22,25,29,34)(23,31,30,33)(27,36,35,32);(5,9,10,18)(6,13,11,17)(7,14,12,19)(8,15,16,20)(21,25)(22,28)(23,32)(24,29)(26,34)(27,31)(30,36)(33,35)Polycyclic presentation:
2^8|a^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 | #132419, #245918, #245919, … |
| maximal normal subgroup of | #132419, #245918, #245919, … |
| maximal quotient of | #560457, #560486, #561553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43880);