Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22,25,28)(23,26,30,34)(24,27,32,36)(29,33,31,35);(6,10)(11,16)(12,17)(18,20)(21,23,25,30)(22,26,28,34)(24,31,32,29)(27,35,36,33);(2,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24)(22,27)(23,29)(25,32)(26,33)(28,36)(30,31)(34,35);(7,13)(11,16)(14,19)(18,20)(22,28)(24,32)(26,34)(29,31);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)Polycyclic presentation:
2^8|a^2=gh,b^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #113086, #113293, #113947, … |
| maximal normal subgroup of | #113086, #113293, #113947, … |
| maximal quotient of | #544271, #544276, #544279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29687);