Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(11,17)(15,20)(18,23)(22,24)(25,26,30,33)(27,29,35,31)(28,36,38,32)(34,39,40,37);(10,14)(11,17)(12,19)(22,24)(25,27,30,35)(26,31,33,29)(28,34,38,40)(32,39,36,37);(1,2,4,5)(3,6,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#45875, #7702426, #7702509, … |
| maximal normal subgroup of | C2×#45875, #7702426, #7702509, … |
| maximal quotient of | #557262, #557328, #557424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38863);