Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
443 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 107 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)(4,6)(5,8)(9,10,14,19)(11,15,16,23)(12,21,22,20)(13,18,24,17);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,20)(14,16)(17,22)(19,23)(21,24);(9,12,14,22)(10,16,19,11)(13,23,24,15)(17,20,18,21);(2,6)(3,8)(9,11)(10,12)(13,21)(14,16)(15,18)(17,23)(19,22)(20,24)Polycyclic presentation:
2^8|a^2=h,c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151339, #151341, #151457, … |
| maximal normal subgroup of | #151339, #151341, #151457, … |
| maximal quotient of | #331669, #331672, #331673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23420);