Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
591 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 591 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,9)(8,11)(10,12)(13,14,16,18)(15,19,20,17)(23,28)(24,26);(6,7)(10,11)(13,14,16,18)(15,17,20,19)(21,22,25,27)(23,26,28,24);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,23)(22,26)(24,27)(25,28);(5,8,9,12)(6,10,7,11)(13,15,16,20)(14,17,18,19)(21,24,25,26)(22,23,27,28)Polycyclic presentation:
2^8|a^2=eg,b^2=eh,c^2=e,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #401720, #401721, #401722, … |
| maximal normal subgroup of | #401720, #401721, #401722, … |
| maximal quotient of | #154187, #154397, #154400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8223);