Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18,20,24,21,23,29,31)(19,25,26,30,27,32,28,22);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16);(11,15)(13,16)(17,19,21,27)(18,22,23,30)(20,28,29,26)(24,32,31,25);(1,4,5,3)(2,7,8,6)(18,23)(22,30)(24,31)(25,32)Polycyclic presentation:
2^8|a^2=efg,b^2=eg,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #100792, #100804, #103859, … |
| maximal normal subgroup of | #100792, #100804, #103859, … |
| maximal quotient of | #326515, #326530, #326531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22135);