Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 65536 |
| 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 | |
1104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,7,8,6)(11,15)(14,16)(18,22)(20,23);(2,5)(6,7)(10,13)(14,16)(18,22)(19,24);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,23,22,20);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,14,13,16)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^8|a^2=f,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6471781, #6471785, #6471788, … |
| maximal normal subgroup of | #6471781, #6471785, #6471788, … |
| maximal quotient of | #553812, #553826, #554049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30319);