Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
956 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)(21,22,27,26)(23,30,25,28)(24,29,34,33)(31,36,32,35);(6,10)(8,16)(11,13)(14,20)(21,23,27,25)(22,28,26,30)(24,32,34,31)(29,36,33,35);(1,2)(3,4)(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18)(21,24,27,34)(22,29,26,33)(23,31,25,32)(28,35,30,36);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,25,27,23)(22,30,26,28)(24,32,34,31)(29,36,33,35);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)(21,26,27,22)(23,30,25,28)(24,33,34,29)(31,36,32,35)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,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#42301, #7640804, #7640806, … |
| maximal normal subgroup of | C2×#42301, #7640804, #7640806, … |
| maximal quotient of | #544483, #545059, #545412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35289);