Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,14,11,13,9,7)(8,16,17,12,18,15,19,20);(5,7,10,13,11,14,9,6)(8,15,19,20,18,16,17,12);(2,4)(5,8)(6,12)(7,15)(9,17)(10,19)(11,18)(13,20)(14,16);(5,9,11,10)(6,7,13,14)(8,17,18,19)(12,15,20,16)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #359619, #359681, #364613, … |
| maximal normal subgroup of | #359619, #359681, #364613, … |
| maximal quotient of | #196174, #196180, #196185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13993);