Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,30,39)(20,32,34,40)(21,26,22,35)(25,36,41,33)(28,42,44,47)(29,38,31,45)(37,46,48,43);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,23,30)(18,24,27,39)(20,28,34,44)(21,29,22,31)(25,37,41,48)(26,38,35,45)(32,42,40,47)(33,43,36,46);(3,10)(6,9)(11,16)(13,15)(17,20)(18,25)(19,28)(21,33)(22,36)(23,34)(24,37)(26,40)(27,41)(29,43)(30,44)(31,46)(32,35)(38,47)(39,48)(42,45);(2,7)(6,9)(8,14)(13,15)(17,21,23,22)(18,26,27,35)(19,29,30,31)(20,33,34,36)(24,38,39,45)(25,40,41,32)(28,43,44,46)(37,47,48,42);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,30)(24,39)(28,44)(29,31)(37,48)(38,45)(42,47)(43,46)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#60987, #10575832, #10575883, … |
| maximal normal subgroup of | C2×#60987, #10575832, #10575883, … |
| maximal quotient of | #6374727, #6374730, #6375096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53975);