Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #427 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1808 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,13)(4,19,21,9)(5,23)(6,26)(7,11)(8,16)(10,14)(12,24)(15,40,42,33)(17,46)(18,35)(20,50,52,53)(22,31,51,49)(25,54)(27,41,47,63)(28,45)(29,56)(30,38,48,60)(32,39)(34,37)(36,44)(43,58,62,61)(55,64,57,59);(1,3,14,37,6,17,25,45,7,18,39,56,24,44,5,16)(2,8,23,36,12,29,32,35,11,28,54,46,26,34,10,13)(4,15,38,59,22,43,52,63,21,42,60,64,51,62,20,41)(9,27,53,58,31,55,48,40,19,47,50,61,49,57,30,33);(1,4)(2,9)(3,15)(5,20)(6,22)(7,21)(8,27)(10,30)(11,19)(12,31)(13,33)(14,38)(16,41)(17,43)(18,42)(23,53)(24,51)(25,52)(26,49)(28,47)(29,55)(32,48)(34,57)(35,40)(36,58)(37,59)(39,60)(44,62)(45,63)(46,61)(50,54)(56,64)Polycyclic presentation:
2^8|a^2=f,b^2=dg,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94218, #94219, #134773, … |
| maximal normal subgroup of | #94218, #94219, #134773, … |
| maximal quotient of | #134675, #134676, #134696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5827);