Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2011 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
489 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 489 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,27)(26,28);(1,2,3,4)(6,9)(10,11)(13,15)(14,18)(16,23)(17,24)(19,26)(20,22)(21,27)(25,28);(2,4)(14,19)(17,25)(18,26)(20,21)(22,27)(24,28);(5,7,8,12)(6,10,9,11)(14,20)(15,23)(19,21)(24,28)Polycyclic presentation:
2^8|b^2=g,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #150477, #150480, #150817, … |
| maximal normal subgroup of | #150477, #150480, #150817, … |
| maximal quotient of | #316817, #316828, #316838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21159);