Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#120 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
889 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,10,14,11,8,9,13)(7,15,17,16,19,20,18,12);(1,2,3,4)(5,7,11,19)(6,12,8,16)(9,17,10,18)(13,15,14,20);(5,8)(6,11)(7,16)(9,14)(10,13)(12,19)(15,18)(17,20);(5,9,11,10)(6,13,8,14)(7,17,19,18)(12,15,16,20)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,e^b=eh,g^b=gh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #354320, #354321, #356680, … |
| maximal normal subgroup of | #354320, #354321, #356680, … |
| maximal quotient of | #193726, #193732, #193741, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13930);