Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) = C26×#442 (order 4096) · Inn(G) = C22×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3280 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
343 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)(5,6,10,15)(7,16,20,27)(8,21,24,30)(9,12,23,14)(11,26,29,18)(13,31,33,22)(17,34,35,28)(19,36,25,32);(5,7,10,20)(6,11,15,29)(8,17,24,35)(9,25,23,19)(12,28,14,34)(13,32,33,36)(16,30,27,21)(18,31,26,22);(2,4);(6,12)(7,17)(8,22)(9,23)(11,27)(14,15)(16,29)(18,19)(20,35)(21,30)(24,31)(25,26)(28,32)(34,36)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gh,d^b=de,e^b=eg,g^b=gh,e^d=eg,g^d=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33832, #7623585, #7623605, … |
| maximal normal subgroup of | C2×#33832, #7623585, #7623605, … |
| maximal quotient of | #491736, #491739, #491746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26820);