Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #62636 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×D8 |
| commutator, Frattini, cosocle | C8 | C25 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1013 conjugacy classes of subgroups; only the 477 normal subgroups are shown.
This group has 1013 conjugacy classes of subgroups — too many to draw. Only its 477 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 — 5 generators on 64 points:
(1,2)(3,13)(4,9)(5,21)(6,26)(7,10)(8,15)(11,24)(12,23)(14,38)(16,45)(17,39)(18,31)(19,48)(20,51)(22,36)(25,37)(27,35)(28,41)(29,44)(30,53)(32,50)(33,47)(34,43)(40,60)(42,63)(46,59)(49,58)(52,57)(54,61)(55,64)(56,62);(1,3,7,17)(2,8,10,29)(4,14,18,40)(5,22,25,53)(6,27,23,16)(9,28,31,54)(11,35,34,45)(12,36,26,30)(13,37,39,21)(15,43,44,24)(19,49,46,64)(20,52,47,42)(32,57,56,63)(33,58,51,55)(38,59,60,48)(41,62,61,50);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,28)(10,31)(11,32)(12,33)(13,38)(15,41)(16,42)(17,40)(21,48)(22,49)(23,47)(24,50)(25,46)(26,51)(27,52)(29,54)(30,55)(34,56)(35,57)(36,58)(37,59)(39,60)(43,62)(44,61)(45,63)(53,64);(4,18)(9,31)(14,40)(19,46)(20,47)(28,54)(32,56)(33,51)(38,60)(41,61)(42,52)(48,59)(49,64)(50,62)(55,58)(57,63);(2,10)(8,29)(9,31)(11,34)(12,26)(13,39)(21,37)(28,54)(30,36)(32,56)(33,51)(35,45)(38,60)(48,59)(55,58)(57,63)Polycyclic presentation:
2^8|b^2=h,f^2=gh,g^2=h,b^a=bf,e^a=eh,f^a=fg,g^a=gh,f^b=fg,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #62637⋊C3 |
| maximal subgroup of | #497431, #499744, #499745, … |
| maximal normal subgroup of | #497431, #499744, #499745, … |
| maximal quotient of | #7622498, #7622499, #7622537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55625);