Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 |
| prime radical | #2661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,20,23,24);(1,2)(3,4)(5,6)(7,8)(18,22)(19,23);(2,6)(3,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,23)(22,24);(1,3,5,7)(2,4,6,8)(17,20)(18,19)(21,24)(22,23);(1,4)(2,7)(3,6)(5,8)(9,11,12,15)(10,13,14,16)Polycyclic presentation:
2^8|a^2=fg,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#42042, #7640316, #7640482, … |
| maximal normal subgroup of | C2×#42042, #7640316, #7640482, … |
| maximal quotient of | #544735, #545223, #545770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35030);