Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C2×#120 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,10)(4,5)(9,12)(11,13)(14,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32);(1,2,6,9)(3,8,11,15)(4,12,5,7)(10,16,13,14)(17,19,22,29)(18,23,25,31)(20,28,21,27)(24,32,30,26);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,15)(12,16)(17,18)(19,23)(20,24)(21,30)(22,25)(26,27)(28,32)(29,31);(2,9)(7,12)(8,15)(14,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)Polycyclic presentation:
2^8|b^2=g,d^2=h,e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19040, #6351443, #6351444, … |
| maximal normal subgroup of | C2×#19040, #6351443, #6351444, … |
| maximal quotient of | #166323, #166333, #166380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12028);