Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
949 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)(6,10)(7,13)(12,17)(14,19)(21,22)(23,30)(24,27)(25,33)(26,29)(28,34)(31,35)(32,36);(6,10)(11,16)(12,17)(18,20)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,24,26,23)(22,27,29,30)(25,32,34,31)(28,35,33,36);(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,26,34)(22,28,29,33)(23,31,24,32)(27,35,30,36)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #107329, #107667, #107817, … |
| maximal normal subgroup of | #107329, #107667, #107817, … |
| maximal quotient of | #513791, #513802, #513803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27391);