Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(17,18)(19,22)(20,21)(23,24);(2,3)(6,7)(9,10,13,11)(12,15,16,14)(17,19,21,24)(18,22,20,23);(10,11)(14,15)(18,20)(22,23);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,20,21,18)(19,23,24,22);(1,4,5,8)(2,6,3,7)(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=gh,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58263, #8104079, #8104087, … |
| maximal normal subgroup of | C2×#58263, #8104079, #8104087, … |
| maximal quotient of | #585468, #586428, #587313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51251);