Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
946 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)(7,8,11,16)(9,12,18,20)(14,15,17,19)(21,22,27,26)(23,30,24,28)(25,33,34,29)(31,35,32,36);(5,7,10,11)(6,8,13,16)(9,17,18,14)(12,19,20,15)(21,23,27,24)(22,28,26,30)(25,31,34,32)(29,35,33,36);(1,2,3,4)(5,7)(6,8)(9,14)(10,11)(12,15)(13,16)(17,18)(19,20)(21,24,27,23)(22,28,26,30)(25,31,34,32)(29,36,33,35);(5,8)(6,11)(7,13)(9,15)(10,16)(12,17)(14,20)(18,19)(21,25,27,34)(22,29,26,33)(23,31,24,32)(28,35,30,36);(5,9,10,18)(6,12,13,20)(7,14,11,17)(8,15,16,19)(21,26,27,22)(23,28,24,30)(25,33,34,29)(31,35,32,36)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=fg,d^2=f,e^2=fh,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 | #132369, #248380, #248381, … |
| maximal normal subgroup of | #132369, #248380, #248381, … |
| maximal quotient of | #544526, #545087, #545501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35879);