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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
928 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)(5,6,10,9)(7,13,15,11)(8,16,18,12)(14,19,17,20);(6,9)(11,13)(12,16)(19,20)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(1,2)(3,4)(5,7,10,15)(6,11,9,13)(8,17,18,14)(12,20,16,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,8,10,18)(6,12,9,16)(7,14,15,17)(11,19,13,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,9,10,6)(7,13,15,11)(8,16,18,12)(14,20,17,19)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,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 | #133239, #214937, #247812, … |
| maximal normal subgroup of | #133239, #214937, #247812, … |
| maximal quotient of | #544451, #545032, #545391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35781);