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 | |
744 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)(3,4)(5,7)(6,8)(11,15)(14,16)(17,18,22,26)(19,23,27,31)(20,24,29,32)(21,28,30,25);(1,3,6,5)(2,7,8,4)(9,10,13,12)(11,14,15,16)(19,27)(21,30)(23,31)(25,28);(9,11)(10,14)(12,16)(13,15)(17,19,22,27)(18,23,26,31)(20,28,29,25)(21,24,30,32);(1,4,6,7)(2,3,8,5)(9,12,13,10)(11,16,15,14)(17,20)(18,24)(19,25)(21,23)(22,29)(26,32)(27,28)(30,31);(1,5,6,3)(2,7,8,4)(17,21)(18,25)(19,24)(20,23)(22,30)(26,28)(27,32)(29,31)Polycyclic presentation:
2^8|a^2=h,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×#44524, #7669066, #7669101, … |
| maximal normal subgroup of | C2×#44524, #7669066, #7669101, … |
| maximal quotient of | #535639, #535704, #536306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37512);