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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
387 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,25,28)(16,21,17,20)(19,26,27,24);(1,3,4,2)(6,9)(7,11);(13,15,18,25)(14,19,22,27)(16,24,17,26)(20,23,21,28);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,23,27,28)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22981, #6381738, #6381742, … |
| maximal normal subgroup of | C2×#22981, #6381738, #6381742, … |
| maximal quotient of | #228837, #228845, #228869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15969);