Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C24 · Out(G) = C211 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,13)(15,16);(2,6)(7,8)(17,18)(19,23)(20,22)(21,24);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,20,24)(18,21,22,23);(1,4,5,3)(2,8,6,7)(18,22)(21,23);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=f,c^2=fh,d^2=f,e^2=g,b^a=bf,c^a=cg,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #6373686, #6373687, #6373688, … |
| maximal normal subgroup of | #6373686, #6373687, #6373688, … |
| maximal quotient of | #514164, #514240, #514394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27872);