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×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(11,15)(14,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30);(2,3)(4,8)(10,13)(11,15)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=fg,e^2=fg,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 | #133541, #500070, #500418, … |
| maximal normal subgroup of | #133541, #500070, #500418, … |
| maximal quotient of | #557145, #557431, #557596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38730);