Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19651 |
| automorphism | Aut(G) order 65536 |
| 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 |
| prime radical | C23 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,22,26)(19,27,31,41)(20,24,32,38)(21,34,28,25)(23,35,37,30)(29,40,43,48)(33,45,42,39)(36,46,47,44);(17,19,28,35,22,31,21,30)(18,23,34,41,26,37,25,27)(20,29,42,46,32,43,33,44)(24,36,45,48,38,47,39,40);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,20,22,32)(18,24,26,38)(19,29,31,43)(21,33,28,42)(23,36,37,47)(25,39,34,45)(27,40,41,48)(30,44,35,46);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(20,32)(24,38)(29,43)(33,42)(36,47)(39,45)(40,48)(44,46)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=fg,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 | #141368, #141372, #141826, … |
| maximal normal subgroup of | #141368, #141372, #141826, … |
| maximal quotient of | #191517, #206145, #206232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12656);