Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,15)(9,10)(12,16)(13,19)(14,17)(18,20)(21,22,26,29)(23,24,31,27)(25,33,34,28)(30,35,36,32);(6,10)(11,17)(12,15)(18,19)(21,23,26,31)(22,27,29,24)(25,30,34,36)(28,35,33,32);(1,2,4,3)(5,7,9,14)(6,11,10,17)(8,13,16,20)(12,18,15,19)(21,23)(22,27)(24,29)(25,30)(26,31)(28,35)(32,33)(34,36);(5,6)(7,11)(8,12)(9,10)(13,18)(14,17)(15,16)(19,20)(21,24,26,27)(22,23,29,31)(25,32,34,35)(28,30,33,36);(1,3,4,2)(5,8)(6,12)(7,13)(9,16)(10,15)(11,18)(14,20)(17,19)(21,25,26,34)(22,28,29,33)(23,30,31,36)(24,32,27,35)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fh,d^2=g,e^2=h,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.
| Fitting subgroup of | #34530⋊C3 |
| maximal subgroup of | #97011, #209160, #209161, … |
| maximal normal subgroup of | #97011, #209160, #209161, … |
| maximal quotient of | #513878, #513967, #513980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27518);