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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
967 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)(7,13)(11,16)(14,19)(18,20)(22,26)(23,29)(24,31)(34,36);(6,10)(7,13)(8,15)(18,20)(21,22,25,26)(23,27,29,33)(24,32,31,28)(30,36,35,34);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,27)(24,28)(25,26)(29,33)(30,34)(31,32)(35,36);(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23)(22,27)(24,30)(25,29)(26,33)(28,34)(31,35)(32,36);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24,25,31)(22,28,26,32)(23,30,29,35)(27,34,33,36)Polycyclic presentation:
2^8|b^2=f,d^2=fh,e^2=h,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×#46277, #7702988, #7702995, … |
| maximal normal subgroup of | C2×#46277, #7702988, #7702995, … |
| maximal quotient of | #557053, #557369, #557647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39265);