Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,9,12)(5,8,13,14)(10,11,15,16)(17,18,23,27)(19,28,24,20)(21,25,35,41)(22,36,37,26)(29,39,33,31)(30,32,38,42)(34,46,47,40)(43,44,45,48);(2,7)(3,9)(8,14)(10,15)(17,19,23,24)(18,20,27,28)(21,33,35,29)(22,30,37,38)(25,39,41,31)(26,32,36,42)(34,45,47,43)(40,48,46,44);(1,3)(2,4)(5,10)(6,9)(7,12)(8,11)(13,15)(14,16)(17,19)(18,20)(21,29)(22,30)(23,24)(25,31)(26,32)(27,28)(33,35)(34,43)(36,42)(37,38)(39,41)(40,44)(45,47)(46,48);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,20)(18,24)(19,27)(21,31)(22,32)(23,28)(25,33)(26,38)(29,41)(30,36)(34,44)(35,39)(37,42)(40,45)(43,46)(47,48);(17,21,23,35)(18,25,27,41)(19,29,24,33)(20,31,28,39)(22,34,37,47)(26,40,36,46)(30,43,38,45)(32,44,42,48);(1,5,6,13)(2,8,7,14)(3,10,9,15)(4,11,12,16)(17,22,23,37)(18,26,27,36)(19,30,24,38)(20,32,28,42)(21,34,35,47)(25,40,41,46)(29,43,33,45)(31,44,39,48)Polycyclic presentation:
2^8|a^2=g,b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh70 conjugacy classes — view the full character table.
| maximal subgroup of | #10580620, #10580622, #10580623, … |
| maximal normal subgroup of | #10580620, #10580622, #10580623, … |
| maximal quotient of | #7626376, #7626384, #7626420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55878);