Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,15)(13,17)(16,20)(18,21)(19,23)(22,24);(2,7)(3,5)(4,8)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22);(1,3,6,5)(2,4,8,7)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23);(1,4,6,7)(2,3,8,5)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=de,e^b=eg,f^c=fh,e^d=eg43 conjugacy classes — view the full character table.
| maximal subgroup of | #491785, #491787, #491801, … |
| maximal normal subgroup of | #491785, #491787, #491801, … |
| maximal quotient of | #220778, #220954, #220978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12896);