Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13523 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2092 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
144 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,11,30,13,34,46,8,15,6,12,7,14,31,47)(3,16,18,50,19,48,32,38,21,52,57,37,20,44,28,9)(4,22,24,42,35,45,33,10,27,40,25,41,26,49,29,43)(17,36,53,58,23,59,60,64,55,62,61,63,54,39,56,51);(1,3,7,19,8,21,30,20)(2,9,14,50,15,38,13,37)(4,23,26,55,27,54,35,17)(5,28,31,18,6,32,34,57)(10,39,42,36,43,59,41,62)(11,44,47,16,12,48,46,52)(22,58,49,64,40,63,45,51)(24,53,29,60,25,61,33,56);(1,4,8,27)(2,10,15,43)(3,17,21,55)(5,29,6,33)(7,35,30,26)(9,36,38,62)(11,45,12,49)(13,42,14,41)(16,51,52,64)(18,56,57,60)(19,54,20,23)(22,47,40,46)(24,34,25,31)(28,53,32,61)(37,59,50,39)(44,58,48,63)Polycyclic presentation:
2^8|a^2=d,b^2=de,c^2=h,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dg,e^b=egh,g^b=gh,d^c=dg,e^c=eg,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13524, #499518, #499604, … |
| maximal normal subgroup of | C2×#13524, #499518, #499604, … |
| maximal quotient of | #146187, #146189, #146201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6512);