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 | |
925 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)(3,7)(4,5)(6,8)(9,10,12,15)(11,14,16,13)(17,18)(19,21)(20,23)(22,24);(2,5)(6,7)(9,11,12,16)(10,13,15,14)(18,19)(22,23);(11,16)(13,14)(17,18,21,19)(20,22,24,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,18)(20,23,24,22);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^8|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=g,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×#56264, #8100615, #8101178, … |
| maximal normal subgroup of | C2×#56264, #8100615, #8101178, … |
| maximal quotient of | #585864, #586191, #586655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49252);