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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,6,7,8)(11,15)(13,16)(17,18)(19,23)(20,22)(21,24);(2,5)(3,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,23)(20,22)(21,24);(1,3,4,7)(2,6,5,8)(9,10,12,14)(11,13,15,16)(17,18,20,22)(19,21,24,23);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=f,c^2=h,e^2=g,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×#44139, #7668165, #7668718, … |
| maximal normal subgroup of | C2×#44139, #7668165, #7668718, … |
| maximal quotient of | #535365, #535681, #536009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37127);