Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,11)(8,13)(9,12)(10,14)(15,18)(16,20)(17,19);(1,2)(3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,18)(14,20);(1,3,4,2)(6,12)(8,9)(11,18)(13,14)(15,17)(19,20);(5,8,10,9)(6,13,14,12)(7,15,16,17)(11,18,20,19)Polycyclic presentation:
2^8|c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #394662, #394663, #394953, … |
| maximal normal subgroup of | #394662, #394663, #394953, … |
| maximal quotient of | #238565, #238568, #238569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16483);