Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,8)(5,7)(10,14,16,20)(15,17,22,23);(1,3,5,4)(2,6,7,8)(9,10)(11,17)(12,20)(13,16)(14,21)(15,19)(18,23)(22,24);(1,4,5,3)(2,6,7,8)(11,18)(15,22)(17,23)(19,24);(9,11,12,19,13,18,21,24)(10,15,14,17,16,22,20,23)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,d^2=e,e^2=h,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #505144, #505145, #505146, … |
| maximal normal subgroup of | #505144, #505145, #505146, … |
| maximal quotient of | #226786, #226787, #226788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15792);