Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) = C28×D4 (order 2048) · Inn(G) = C22×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | #3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,13,10,7,14,12)(8,11,17,19,16,15,20,18);(1,2,3,4)(8,16)(11,15)(17,20)(18,19);(5,7)(6,10)(8,15)(9,12)(11,16)(13,14)(17,18)(19,20);(5,8,9,17,10,16,14,20)(6,11,13,19,7,15,12,18)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=g,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21519, #6372838, #6372839, … |
| maximal normal subgroup of | C2×#21519, #6372838, #6372839, … |
| maximal quotient of | #210452, #210453, #210454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14507);