Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×Q8 | C24 |
| prime radical | #3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
588 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 181 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,3,13)(4,9,18,25,16,23,14,27)(5,10,15,22,20,26,19,8)(17,24,28,31,29,32,30,21);(1,3,7,6)(2,8,12,22)(4,14,16,18)(5,19,20,15)(9,21,23,31)(10,11,26,13)(17,30,29,28)(24,25,32,27);(1,4)(2,9)(3,14)(5,17)(6,18)(7,16)(8,21)(10,24)(11,25)(12,23)(13,27)(15,28)(19,30)(20,29)(22,31)(26,32);(4,16)(9,23)(14,18)(17,29)(21,31)(24,32)(25,27)(28,30)Polycyclic presentation:
2^8|a^2=f,b^2=h,e^2=h,f^2=h,b^a=be,e^a=eg,g^a=gh,e^b=eh,f^b=fgh,d^c=dh,f^e=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #151824, #151831, #152706, … |
| maximal normal subgroup of | #151824, #151831, #152706, … |
| maximal quotient of | #480016, #480048, #480116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26360);