Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(10,14)(15,21)(16,22)(23,24)(25,27,29,34)(26,30,32,38)(28,35,36,33)(31,39,40,37);(2,5)(6,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(26,32)(28,36)(30,38)(33,35);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #36701⋊C3 |
| maximal subgroup of | #97007, #241605, #241608, … |
| maximal normal subgroup of | #97007, #241605, #241608, … |
| maximal quotient of | #544327, C4×#2822, C4×#2845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29689);