Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2504 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 109 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 — 4 generators on 64 points:
(1,2,8,12,7,14,6,13)(3,15,19,32,20,39,18,9)(4,21,23,36,24,35,28,10)(5,25,29,38,27,37,26,11)(16,33,43,40,45,30,44,57)(17,46,48,59,49,31,47,41)(22,34,51,50,52,58,54,53)(42,61,62,56,64,60,63,55);(1,3,6,18,7,20,8,19)(2,9,12,15,14,32,13,39)(4,16,23,43,24,45,28,44)(5,17,26,47,27,49,29,48)(10,33,35,57,36,30,21,40)(11,31,25,41,38,46,37,59)(22,42,51,62,52,64,54,63)(34,56,53,61,58,55,50,60);(1,4,7,24)(2,10)(3,16)(5,22,27,52)(6,23,8,28)(9,30,32,33)(11,34)(12,35)(13,21)(14,36)(15,40,39,57)(17,42)(18,43)(19,44)(20,45)(25,53)(26,51,29,54)(31,55,46,56)(37,50)(38,58)(41,60,59,61)(47,62)(48,63)(49,64);(1,5,8,29,7,27,6,26)(2,11,13,37,14,38,12,25)(3,17,19,48,20,49,18,47)(4,22,28,54,24,52,23,51)(9,31,39,59,32,46,15,41)(10,34,21,50,36,58,35,53)(16,42,44,63,45,64,43,62)(30,55,57,61,33,56,40,60)Polycyclic presentation:
2^8|a^2=eg,b^2=e,c^2=ef,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142697, #142699, #142701, … |
| maximal normal subgroup of | #142697, #142699, #142701, … |
| maximal quotient of | #316150, #316152, #316155, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20976);