Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 262144 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(8,16)(11,18)(14,20)(17,19)(22,24)(23,29)(25,33)(34,36);(5,6,9,12)(7,10,15,13)(8,11,16,18)(14,17,20,19)(21,22,26,24)(23,27,29,31)(25,28,33,32)(30,34,35,36);(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,36)(33,35);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,24,26,22)(23,27,29,31)(25,32,33,28)(30,34,35,36);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^8|a^2=fg,b^2=fh,d^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #111988, #112301, #113066, … |
| maximal normal subgroup of | #111988, #112301, #113066, … |
| maximal quotient of | #556357, #556358, #556374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30536);