Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
658 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)(5,6,10,16,12,8,11,15)(7,17,21,18,22,29,20,13)(9,23,26,32,27,14,25,31)(19,30,24,28,36,33,35,34);(5,7)(6,13)(8,18)(9,24)(10,21)(11,20)(12,22)(14,30)(15,29)(16,17)(19,25)(23,33)(26,36)(27,35)(28,31)(32,34);(2,4)(5,8,12,6)(7,18,22,13)(9,23,27,14)(10,16,11,15)(17,20,29,21)(19,34,36,28)(24,33,35,30)(25,32,26,31);(5,9)(6,14)(7,19)(8,23)(10,26)(11,25)(12,27)(13,28)(15,32)(16,31)(17,33)(18,34)(20,35)(21,24)(22,36)(29,30)Polycyclic presentation:
2^8|a^2=g,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,d^b=de,e^b=eh,f^c=fh,g^c=gh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #497460, #497461, #497463, … |
| maximal normal subgroup of | #497460, #497461, #497463, … |
| maximal quotient of | #220225, #220231, #220235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15456);