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 | |
989 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)(3,5)(4,6)(7,8)(17,18)(19,21)(20,23)(22,24);(4,8)(6,7)(9,10,13,14)(11,12,15,16)(18,19)(22,23);(2,3)(6,7)(10,14)(11,15)(17,18,21,19)(20,22,24,23);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,15)(14,16)(17,19,21,18)(20,23,24,22);(1,4,5,8)(2,6,3,7)(9,12)(10,11)(13,16)(14,15)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^8|b^2=fh,c^2=f,d^2=g,e^2=fg,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×#55434, #8098813, #8098845, … |
| maximal normal subgroup of | C2×#55434, #8098813, #8098845, … |
| maximal quotient of | #585803, #586345, #586835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48422);