Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C23 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,15,20,16,21,14,31)(11,17,25,33,26,34,24,39)(12,27,13,19,30,36,29,18)(22,37,38,35,28,40,23,32);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,28)(13,23)(14,24)(15,25)(16,26)(18,35)(19,32)(20,33)(21,34)(22,30)(27,40)(29,38)(31,39)(36,37);(9,12,14,29,16,30,15,13)(10,18,20,27,21,19,31,36)(11,22,24,23,26,28,25,38)(17,32,33,37,34,35,39,40);(1,4,5,3)(2,6,7,8)(9,13,15,30,16,29,14,12)(10,19,20,36,21,18,31,27)(11,23,25,22,26,38,24,28)(17,32,33,37,34,35,39,40)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=eg,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #125059, #125060, #136780, … |
| maximal normal subgroup of | #125059, #125060, #136780, … |
| maximal quotient of | C8×#501, #213409, #213414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14723);