Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(3,7)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(1,2,5,4)(3,8,7,6)(10,14)(11,18)(16,21)(19,24);(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,3,5,7)(2,6,4,8)(9,11,13,18)(10,15,14,17)(12,19,20,24)(16,22,21,23)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,4,5,2)(3,8,7,6)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=g,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52828, #8092636, #8092703, … |
| maximal normal subgroup of | C2×#52828, #8092636, #8092703, … |
| maximal quotient of | #582346, #582503, #582559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45816);