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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
985 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)(8,16)(12,17)(14,20)(18,19)(22,23)(24,31)(25,33)(34,35);(6,10)(11,13)(12,17)(18,19)(21,22,26,23)(24,27,31,29)(25,28,33,30)(32,34,36,35);(1,2)(3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,23,26,22)(24,27,31,29)(25,30,33,28)(32,34,36,35);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24,26,31)(22,27,23,29)(25,32,33,36)(28,34,30,35);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=fh,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 | #111984, #113072, #113403, … |
| maximal normal subgroup of | #111984, #113072, #113403, … |
| maximal quotient of | #556342, #556343, #556376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30595);