Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | Aut(G) order 32768 |
| 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; only the 115 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,11)(4,7)(5,15)(6,10)(8,13)(9,12)(14,16)(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);(1,3,12,14,6,13,5,4)(2,7,15,8,10,16,9,11)(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);(1,4,5,13,6,14,12,3)(2,8,9,7,10,11,15,16)(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);(3,13)(4,14)(7,16)(8,11)(20,32)(24,38)(29,43)(33,42)(36,47)(39,45)(40,48)(44,46)Polycyclic presentation:
2^8|a^2=gh,b^2=eg,c^2=fgh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dh,d^c=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | #143169, C2×#19732, #6357371, … |
| maximal normal subgroup of | #143169, C2×#19732, #6357371, … |
| maximal quotient of | #191526, #206263, #206273, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12720);