Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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 | C22×C4⋊C4 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
380 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)(4,5)(7,8)(9,10,13,17,14,16,12,21)(11,18,19,15,20,24,22,23);(9,11,13,19,14,20,12,22)(10,15,17,24,16,23,21,18);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,16)(15,23)(17,21)(18,24)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #109758, #109759, #109762, … |
| maximal normal subgroup of | #109758, #109759, #109762, … |
| maximal quotient of | #207297, #207298, #207299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14359);