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) = C26×C4 |
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 | |
1038 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)(9,10)(11,15)(12,13)(14,16);(4,8)(6,7)(9,11,13,16)(10,14,12,15)(17,18,21,23)(19,20,24,22);(2,3)(6,7)(11,16)(14,15)(17,19,21,24)(18,22,23,20);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,24)(22,23);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,14,16,15)(17,20)(18,19)(21,22)(23,24)Polycyclic presentation:
2^8|b^2=h,c^2=fh,d^2=g,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×#57193, #8100823, #8101177, … |
| maximal normal subgroup of | C2×#57193, #8100823, #8101177, … |
| maximal quotient of | #585329, #585555, #585944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50181);