Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(4,7)(10,13)(14,16)(18,23)(19,24);(2,6)(3,8)(17,18,22,23)(19,20,24,21);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,23)(22,24);(1,3)(2,7)(4,6)(5,8)(17,20,22,21)(18,24,23,19);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,21,22,20)(18,19,23,24)Polycyclic presentation:
2^8|b^2=f,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#35134, #7628348, #7628361, … |
| maximal normal subgroup of | C2×#35134, #7628348, #7628361, … |
| maximal quotient of | #516500, #516503, #516678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28122);