Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,4)(6,7)(11,15)(14,16)(17,18,22,23)(19,20,25,28)(21,29,30,24)(26,32,31,27);(1,2,5,8)(3,7,4,6)(10,13)(11,15)(18,23)(19,25)(21,30)(27,32);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,32)(30,31);(1,4,5,3)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,31)(30,32);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,30)(18,24,23,29)(19,26,25,31)(20,27,28,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,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 | #7206480, #7206484, #7206487, … |
| maximal normal subgroup of | #7206480, #7206484, #7206487, … |
| maximal quotient of | #557241, #557340, #557424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38636);