Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,15)(4,20)(5,11)(6,14)(7,13)(8,12)(9,18)(10,22)(16,30)(17,42)(19,38)(21,50)(23,40)(24,55)(25,39)(26,36)(27,32)(28,37)(29,34)(31,47)(33,53)(35,48)(41,45)(43,56)(44,49)(46,57)(51,61)(52,59)(54,60)(58,63)(62,64);(1,3)(2,9)(4,16)(5,24)(6,27)(7,18)(8,19)(10,30)(11,35)(12,32)(13,38)(14,15)(17,25)(20,49)(21,51)(22,45)(23,46)(26,47)(28,48)(29,44)(31,37)(33,58)(34,57)(36,42)(39,55)(40,41)(43,52)(50,64)(53,62)(54,63)(56,60)(59,61);(1,4,8,23)(2,10,12,34)(3,16,19,46)(5,21,26,54)(6,22,7,29)(9,30,32,57)(11,33,36,59)(13,20,14,40)(15,41,38,49)(17,43,48,62)(18,44,27,45)(24,51,47,63)(25,52,28,53)(31,56,55,64)(35,58,42,61)(37,60,39,50);(1,5)(2,11)(3,17)(4,21)(6,28)(7,25)(8,26)(9,31)(10,33)(12,36)(13,39)(14,37)(15,42)(16,43)(18,47)(19,48)(20,50)(22,53)(23,54)(24,27)(29,52)(30,56)(32,55)(34,59)(35,38)(40,60)(41,61)(44,63)(45,51)(46,62)(49,58)(57,64);(2,12)(9,32)(10,34)(11,36)(13,14)(15,38)(20,40)(30,57)(31,55)(33,59)(35,42)(37,39)(41,49)(50,60)(56,64)(58,61)Polycyclic presentation:
2^8|c^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gh,d^b=df,f^b=fh,g^c=gh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #499778, #499779, #499782, … |
| maximal normal subgroup of | #499778, #499779, #499782, … |
| maximal quotient of | #6358428, #6358430, #6358434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54096);