Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 131072 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
935 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)(5,6,10,12)(7,13,14,8)(9,17,18,11)(15,16,20,19)(21,22,26,29)(23,30,24,27)(25,33,34,28)(31,35,32,36);(7,14)(8,13)(15,20)(16,19)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(1,2)(3,4)(5,7)(6,8)(9,15)(10,14)(11,16)(12,13)(17,19)(18,20)(21,24,26,23)(22,27,29,30)(25,32,34,31)(28,35,33,36);(5,8)(6,7)(9,16)(10,13)(11,15)(12,14)(17,20)(18,19)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34);(5,9,10,18)(6,11,12,17)(7,15,14,20)(8,16,13,19)(21,25,26,34)(22,28,29,33)(23,31,24,32)(27,35,30,36)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6481727, #6481729, #6481731, … |
| maximal normal subgroup of | #6481727, #6481729, #6481731, … |
| maximal quotient of | #553723, #553980, #554201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30365);