Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,8,5,7)(9,10,13,16)(11,14,12,15)(17,18,22,23)(19,27,28,24)(20,25,30,29)(21,31,32,26);(1,3,6,5)(2,7,4,8)(9,11,13,12)(10,14,16,15)(18,23)(19,28)(20,30)(21,32);(9,10)(11,14)(12,15)(13,16)(17,19,22,28)(18,24,23,27)(20,31,30,26)(21,29,32,25);(1,4,6,2)(3,7,5,8)(9,12,13,11)(10,15,16,14)(17,20)(18,25)(19,26)(21,27)(22,30)(23,29)(24,32)(28,31);(1,5,6,3)(2,7,4,8)(17,21,22,32)(18,26,23,31)(19,29,28,25)(20,27,30,24)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#44541, #7669078, #7669102, … |
| maximal normal subgroup of | C2×#44541, #7669078, #7669102, … |
| maximal quotient of | #535642, #535704, #536333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37529);