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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
780 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,25)(19,23,26,31)(20,21,24,30)(27,32,29,28);(1,3,6,4)(2,7,5,8)(10,12)(14,15)(19,26)(20,24)(21,30)(23,31);(3,4)(7,8)(9,10,13,12)(11,15,16,14)(17,19)(18,23)(20,29)(21,28)(22,26)(24,27)(25,31)(30,32);(1,4,6,3)(2,7,5,8)(9,11,13,16)(10,14,12,15)(17,20,22,24)(18,21,25,30)(19,27,26,29)(23,32,31,28);(1,5,6,2)(3,7,4,8)(9,12,13,10)(11,15,16,14)(17,21)(18,24)(19,28)(20,25)(22,30)(23,27)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^2=g,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 | #132473, #253520, #253522, … |
| maximal normal subgroup of | #132473, #253520, #253522, … |
| maximal quotient of | #560401, #560514, #561573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43905);