Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
991 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,11)(8,16)(9,10)(12,17)(13,14)(15,20)(18,19)(21,22,26,27)(23,29,31,24)(25,33,34,28)(30,32,36,35);(6,10)(7,13)(8,17)(18,20)(22,27)(24,29)(28,33)(32,35);(1,2)(3,4)(5,6)(7,14)(8,12)(9,10)(11,13)(15,20)(16,17)(18,19)(21,23)(22,24)(25,30)(26,31)(27,29)(28,32)(33,35)(34,36);(5,7)(6,11)(8,15)(9,13)(10,14)(12,18)(16,20)(17,19)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(5,8,9,17)(6,12,10,16)(7,15,13,19)(11,18,14,20)(21,25,26,34)(22,28,27,33)(23,30,31,36)(24,32,29,35)Polycyclic presentation:
2^8|a^2=fh,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46822, #7700788, #7700792, … |
| maximal normal subgroup of | C2×#46822, #7700788, #7700792, … |
| maximal quotient of | #576625, #576669, #576734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39810);