Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,32)(28,36)(29,34)(30,31)(33,39)(35,38)(37,40);(1,2,4,6)(3,5,7,8)(10,11)(15,17)(16,18)(22,23)(26,31)(29,32)(33,36)(38,40);(1,3,4,7)(2,5,6,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27,30,34)(26,32,31,29)(28,37,39,35)(33,38,36,40);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,39)(26,33,31,36)(27,35,34,37)(29,38,32,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,32)(26,34,31,27)(28,38,39,40)(33,37,36,35)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fg,d^2=f,e^2=f,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 | #132446, #253862, #253864, … |
| maximal normal subgroup of | #132446, #253862, #253864, … |
| maximal quotient of | #544547, #545123, #545409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34428);