Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | Aut(G) = C23×#72236189 (order 8192) · Inn(G) = #392 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(5,6)(7,13)(8,17)(9,12)(10,15)(11,18)(14,20)(16,19);(5,7,8,15,9,16,18,20)(6,10,11,13,12,14,17,19);(5,6)(7,14,16,10)(8,11)(9,12)(13,15,19,20)(17,18)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=e,d^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11333, #358481, C4×#2008, … |
| maximal normal subgroup of | C2×#11333, #358481, C4×#2008, … |
| maximal quotient of | #121363, #121364, #121376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4321);