Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33348 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#99 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| prime radical | #3485 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,8)(5,17,18,9)(6,10)(7,13,20,27)(12,23)(14,24)(15,21)(16,29,28,22)(19,26,30,32)(25,31);(1,3,6,14)(2,7,10,20)(4,12,15,25)(5,13,18,27)(8,19,21,30)(9,11,17,24)(16,26,28,32)(22,23,29,31);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,22)(10,21)(11,23)(13,26)(14,25)(17,29)(18,28)(20,30)(24,31)(27,32);(4,15)(8,21)(12,25)(16,28)(19,30)(22,29)(23,31)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,b^a=be,e^a=eg,g^a=gh,f^b=fg,d^c=dh,f^e=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #493475, #493476, #493477, … |
| maximal normal subgroup of | #493475, #493476, #493477, … |
| maximal quotient of | #480012, #480050, #480097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26338);