Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
729 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)(5,6,10,14,11,15,9,24)(7,16,20,26,18,30,19,12)(8,13,23,28,21,27,22,34)(17,29,32,35,33,36,31,25);(2,4)(5,7)(6,12,15,26)(8,17)(9,19)(10,20)(11,18)(13,25,27,35)(14,16,24,30)(21,33)(22,31)(23,32)(28,29,34,36);(5,8,10,23,11,21,9,22)(6,13,14,28,15,27,24,34)(7,17,20,32,18,33,19,31)(12,25,16,29,26,35,30,36);(7,18)(8,21)(12,26)(13,27)(16,30)(19,20)(22,23)(28,34)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,e^2=h,f^2=h,b^a=be,e^a=eh,g^a=gh,d^b=dh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #355850, #355851, #355852, … |
| maximal normal subgroup of | #355850, #355851, #355852, … |
| maximal quotient of | #182293, #182297, #182300, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13410);