Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2433 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
701 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 701 conjugacy classes of subgroups — too many to draw. Only its 141 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:
(1,2,7,12)(3,13,17,38)(4,9,21,33)(5,22,25,52)(6,26,23,11)(8,27,30,16)(10,34,35,24)(14,39,18,36)(15,43,45,29)(19,48,50,64)(20,51,40,32)(28,42,55,47)(31,56,57,49)(37,53,59,44)(41,54,46,62)(58,61,60,63);(1,3,6,16,7,17,23,27)(2,8,11,13,12,30,26,38)(4,18,20,47,21,14,40,42)(5,15,24,44,25,45,34,53)(9,28,32,36,33,55,51,39)(10,29,22,37,35,43,52,59)(19,46,49,63,50,41,56,61)(31,54,48,58,57,62,64,60);(1,4)(2,9)(3,14,17,18)(5,19)(6,20)(7,21)(8,28,30,55)(10,31)(11,32)(12,33)(13,36,38,39)(15,41,45,46)(16,42,27,47)(22,48)(23,40)(24,49)(25,50)(26,51)(29,54,43,62)(34,56)(35,57)(37,58,59,60)(44,61,53,63)(52,64);(1,5,23,34,7,25,6,24)(2,10,26,52,12,35,11,22)(3,15,27,53,17,45,16,44)(4,19,40,56,21,50,20,49)(8,29,38,59,30,43,13,37)(9,31,51,64,33,57,32,48)(14,41,47,63,18,46,42,61)(28,54,39,60,55,62,36,58)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,d^2=eh,e^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 | #142617, #142619, #142621, … |
| maximal normal subgroup of | #142617, #142619, #142621, … |
| maximal quotient of | #299196, #299199, #299201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18788);