Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| 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 | |
3458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,15,19)(11,20,21,16)(12,17,25,31)(13,14,18,28)(22,33,35,29)(23,34,30,24)(26,27,32,38)(36,40,39,37);(2,6)(3,7)(11,21)(12,25)(16,20)(17,31)(23,30)(24,34)(26,32)(27,38);(1,3)(2,4)(5,7)(6,8)(9,11,15,21)(10,16,19,20)(12,22,25,35)(13,23,18,30)(14,24,28,34)(17,29,31,33)(26,36,32,39)(27,37,38,40);(1,4)(2,3)(5,8)(6,7)(9,12)(10,17)(11,22)(13,26)(14,27)(15,25)(16,29)(18,32)(19,31)(20,33)(21,35)(23,36)(24,37)(28,38)(30,39)(34,40);(9,13)(10,14)(11,23)(12,26)(15,18)(16,24)(17,27)(19,28)(20,34)(21,30)(22,36)(25,32)(29,37)(31,38)(33,40)(35,39);(9,14)(10,18)(11,24)(12,27)(13,19)(15,28)(16,30)(17,32)(20,23)(21,34)(22,37)(25,38)(26,31)(29,39)(33,36)(35,40)Polycyclic presentation:
2^8|a^2=gh,c^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh70 conjugacy classes — view the full character table.
| maximal subgroup of | #347437, #347457, #347478, … |
| maximal normal subgroup of | #347437, #347457, #347478, … |
| maximal quotient of | #7627418, #7627420, #7627494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56051);