Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
871 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 871 conjugacy classes of subgroups — too many to draw. Only its 287 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)(5,6,7,8)(14,18)(19,21)(20,26)(27,28);(9,10,11,12)(13,14,17,18)(15,21,23,19)(16,20,25,26)(22,28,24,27);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,17,23)(14,19,18,21)(16,24,25,22)(20,28,26,27);(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #111138, #112074, #112078, … |
| maximal normal subgroup of | #111138, #112074, #112078, … |
| maximal quotient of | #153907, #153919, #153939, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7262);