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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
959 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)(21,22,26,29)(23,24,27,32)(25,28,34,36)(30,31,33,35);(6,10)(7,13)(8,17)(18,20)(21,23,26,27)(22,24,29,32)(25,33,34,30)(28,35,36,31);(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)(26,27)(28,31)(29,32)(33,34)(35,36);(5,7,9,13)(6,11,10,14)(8,15,17,19)(12,18,16,20)(21,24)(22,27)(23,29)(25,31)(26,32)(28,33)(30,36)(34,35);(5,8)(6,12)(7,15)(9,17)(10,16)(11,18)(13,19)(14,20)(21,25)(22,28)(23,30)(24,31)(26,34)(27,33)(29,36)(32,35)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=g,d^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×#47206, #7700916, #7700944, … |
| maximal normal subgroup of | C2×#47206, #7700916, #7700944, … |
| maximal quotient of | #576620, #576667, #576690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40194);