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 | #3500 | 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 | |
604 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 604 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:
(1,2,5,10,6,11,18,25)(3,8,14,23,12,21,27,31)(4,9,16,24,15,7,17,20)(13,22,28,32,26,19,29,30);(2,7)(4,15)(8,19)(9,11)(10,24)(13,26)(16,17)(20,25)(21,22)(23,32)(28,29)(30,31);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,22)(10,23)(11,21)(15,26)(16,28)(17,29)(18,27)(20,30)(24,32)(25,31);(3,12)(8,21)(13,26)(14,27)(19,22)(23,31)(28,29)(30,32)Polycyclic presentation:
2^8|a^2=f,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 | #151825, #151830, #152704, … |
| maximal normal subgroup of | #151825, #151830, #152704, … |
| maximal quotient of | #480014, #480046, #480117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26358);