Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3036 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,24)(20,22)(21,23);(2,5)(6,7)(9,11,12,19,14,20,13,21)(10,15,16,24,18,22,17,23);(10,16)(11,19)(12,13)(15,22)(17,18)(20,21);(1,3,4,8)(2,6,5,7)(11,20)(15,22)(19,21)(23,24)Polycyclic presentation:
2^8|b^2=gh,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #332330, #332333, #332336, … |
| maximal normal subgroup of | #332330, #332333, #332336, … |
| maximal quotient of | #316813, #316814, #316818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21166);