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 | #2404 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(11,15)(14,16)(17,18,23,25)(19,26,20,21)(22,30,31,24)(27,29,28,32);(2,3)(4,8)(10,13)(11,15)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,30,32);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,21,25,26)(22,28,31,27)(24,29,30,32);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,30)(31,32);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,22)(18,24)(19,27)(20,28)(21,29)(23,31)(25,30)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=fg,d^2=g,e^2=g,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 | #133538, #500068, #500416, … |
| maximal normal subgroup of | #133538, #500068, #500416, … |
| maximal quotient of | #557175, #557469, #557583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38719);