Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
372 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,6)(7,8)(9,10)(11,14)(12,15)(13,17)(16,20)(18,22)(19,21)(23,24);(1,2,6,8,4,7,3,5)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24);(10,15)(12,20)(14,21)(16,17)(19,24)(22,23);(2,7)(5,8)(11,18)(14,22)(19,24)(21,23)Polycyclic presentation:
2^8|b^2=eg,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dgh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #268043, #268049, #268059, … |
| maximal normal subgroup of | #268043, #268049, #268059, … |
| maximal quotient of | #189151, #197880, #197965, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12446);