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:
(2,4)(3,7)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2,5,4)(3,8,7,6)(10,14)(11,18)(16,21)(19,24)(26,28)(31,33)(32,36)(38,39);(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4,5,2)(3,8,7,6)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=g,d^2=g,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×#52852, #8092684, #8092704, … |
| maximal normal subgroup of | C2×#52852, #8092684, #8092704, … |
| maximal quotient of | #582322, #582525, #582551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45840);