Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,14,11,16,10,15)(7,12,18,27,20,29,19,28)(8,21,25,13,24,32,23,31)(17,33,22,26,35,30,34,36);(2,4)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,30)(14,27)(15,28)(16,29)(17,23)(21,26)(24,34)(25,35)(31,33)(32,36);(5,8)(6,13)(7,17)(9,23)(10,25)(11,24)(12,26)(14,21)(15,32)(16,31)(18,34)(19,22)(20,35)(27,33)(28,30)(29,36);(2,4)(5,9,11,10)(6,14,16,15)(7,18,20,19)(8,23,24,25)(12,27,29,28)(13,21,31,32)(17,34,35,22)(26,33,36,30)Polycyclic presentation:
2^8|a^2=fh,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24403, #6437792, #6437795, … |
| maximal normal subgroup of | C2×#24403, #6437792, #6437795, … |
| maximal quotient of | #267056, #267061, #267068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17391);