Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22284 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2328 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 149 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)(3,14)(4,20,24,34)(5,25)(6,12)(7,13)(8,17)(9,23,35,49)(10,27)(11,26)(15,40,43,29)(16,44)(18,31)(19,38)(21,32,42,57)(22,33,52,58)(28,36)(30,47)(37,54)(39,45)(41,61,63,56)(46,59)(48,64,53,51)(50,60,62,55);(1,3,6,17,7,19,26,18)(2,8,11,14,13,31,12,38)(4,21,49,15,24,42,23,43)(5,16,27,45,28,47,54,46)(9,32,20,29,35,57,34,40)(10,30,25,39,37,44,36,59)(22,50,64,41,52,62,51,63)(33,56,48,55,58,61,53,60);(1,4,7,24)(2,9)(3,15)(5,22,28,52)(6,23,26,49)(8,29,31,40)(10,33)(11,34)(12,20)(13,35)(14,32,38,57)(16,41)(17,21)(18,42)(19,43)(25,53)(27,51,54,64)(30,55,44,60)(36,48)(37,58)(39,56,59,61)(45,50)(46,62)(47,63);(1,5,26,54,7,28,6,27)(2,10,12,36,13,37,11,25)(3,16,18,46,19,47,17,45)(4,22,49,64,24,52,23,51)(8,30,38,59,31,44,14,39)(9,33,20,48,35,58,34,53)(15,41,42,62,43,63,21,50)(29,55,57,61,40,60,32,56)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=ef,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #496131, #496132, #497148, … |
| maximal normal subgroup of | #496131, #496132, #497148, … |
| maximal quotient of | #219486, #219488, #219489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15273);