Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19651 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9)(3,10,8,5)(4,7,12,14)(11,16,15,13)(17,18)(19,26)(20,25)(21,23)(22,30)(24,29)(27,31)(28,32);(1,3,5,2,6,8,10,9)(4,11,13,7,12,15,16,14)(18,23)(24,31)(25,30)(26,32);(1,4)(2,7)(3,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,19,22,29,21,28,20,27)(18,24,25,26,23,31,30,32);(4,12)(7,14)(11,15)(13,16)(17,20,21,22)(18,25,23,30)(19,27,28,29)(24,26,31,32)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=f,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^c=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #219652, #219665, #219672, … |
| maximal normal subgroup of | #219652, #219665, #219672, … |
| maximal quotient of | #191520, #206143, #206144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12658);