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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(25,26,31,30)(27,33,28,32)(29,34,38,37)(35,40,36,39);(1,2)(3,6)(4,5)(7,8)(10,11)(12,19)(16,18)(20,24)(25,27,31,28)(26,32,30,33)(29,36,38,35)(34,40,37,39);(2,5)(6,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28,31,27)(26,33,30,32)(29,36,38,35)(34,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,26)(27,33,28,32)(29,37,38,34)(35,40,36,39)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,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×#52525, #8092310, #8092468, … |
| maximal normal subgroup of | C2×#52525, #8092310, #8092468, … |
| maximal quotient of | #582342, #582505, #582553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45513);