Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,14,11,15,9,25)(7,16,20,27,18,30,19,12)(8,21,23,28,24,35,22,13)(17,31,34,36,32,26,33,29);(1,2,3,4)(5,7)(6,12,15,27)(8,17)(9,19)(10,20)(11,18)(13,26,28,31)(14,16,25,30)(21,29,35,36)(22,33)(23,34)(24,32);(5,8,9,22,11,24,10,23)(6,13,14,21,15,28,25,35)(7,17,19,33,18,32,20,34)(12,26,16,29,27,31,30,36);(7,18)(12,27)(16,30)(17,32)(19,20)(26,31)(29,36)(33,34)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124262, #140620, #147407, … |
| maximal normal subgroup of | #124262, #140620, #147407, … |
| maximal quotient of | #204262, #204263, #204264, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14228);