Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4)(6,7)(9,10)(11,19)(12,20)(13,16)(14,18)(15,21)(17,23)(22,24);(5,6,8,7)(9,11,14,15,13,20,23,24)(10,12,18,22,16,19,17,21);(5,7,8,6)(9,12,13,19)(10,15,16,24)(11,18,20,17)(14,22,23,21)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,e^c=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10809, #353511, #353512, … |
| maximal normal subgroup of | C2×#10809, #353511, #353512, … |
| maximal quotient of | #108422, #108423, #108424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3797);