Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 524288 |
| 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 | |
953 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)(11,18)(14,20)(17,19)(21,22,26,24)(23,29,31,27)(25,32,33,28)(30,34,36,35);(5,6,9,12)(7,10,15,13)(8,11,16,18)(14,17,20,19)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36);(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,24,26,22)(23,29,31,27)(25,28,33,32)(30,34,36,35);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,d^2=f,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 | #111980, #112042, #112060, … |
| maximal normal subgroup of | #111980, #112042, #112060, … |
| maximal quotient of | #556356, #556366, #556392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30528);