Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(9,10,14,12)(11,17,18,15)(13,16,21,20)(19,23,24,22)(26,27)(28,35)(32,34)(36,40);(11,18)(13,21)(15,17)(16,20)(25,26,30,27)(28,31,35,33)(29,34,37,32)(36,39,40,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39);(1,3,5,4)(2,7,6,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41786, #7640031, #7640243, … |
| maximal normal subgroup of | C2×#41786, #7640031, #7640243, … |
| maximal quotient of | #544965, #545336, #545981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34774);