Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #2631 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(9,10,14,15)(11,16,13,12)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32);(1,3,6,4)(2,7,8,5)(10,15)(11,13)(18,22)(23,25)(24,28)(30,31);(2,8)(3,4)(9,11,14,13)(10,16,15,12)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,4,6,3)(2,7,8,5)(9,12,14,16)(10,11,15,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,5,6,7)(2,3,8,4)(9,13,14,11)(10,16,15,12)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132485, #253524, #253555, … |
| maximal normal subgroup of | #132485, #253524, #253555, … |
| maximal quotient of | #560387, #560514, #561572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44218);