Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,19,20,18,15,16,17);(5,7,9,8)(6,10,11,12)(13,15,18,14)(16,17,19,20);(5,8,9,7)(6,10,11,12)(13,16,18,19)(14,20,15,17);(13,17,16,15,18,20,19,14)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,d^2=e,e^2=h,g^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 | C2×#22817, #6380648, #6380656, … |
| maximal normal subgroup of | C2×#22817, #6380648, #6380656, … |
| maximal quotient of | #226786, #226803, #226807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15805);