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 | |
703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(10,14)(15,18)(16,21)(22,24)(25,26,30,27)(28,31,36,33)(29,34,37,32)(35,39,40,38);(1,3,5,2)(4,7,8,6)(10,14)(11,17)(12,20)(22,24)(26,27)(29,37)(31,33)(35,40);(1,4)(2,6)(3,7)(5,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^8|a^2=gh,b^2=fgh,c^2=f,d^2=h,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 | #132495, #253739, #253754, … |
| maximal normal subgroup of | #132495, #253739, #253754, … |
| maximal quotient of | #560490, #560617, #561502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44184);