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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
351 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)(7,15)(8,14)(9,23)(11,12)(13,18)(16,28)(17,29)(19,32)(21,22)(24,26)(27,31)(30,36)(33,35);(5,7,12,17)(6,13,11,26)(8,16,22,30)(9,24,23,18)(10,15,25,29)(14,27,21,35)(19,33,32,31)(20,28,34,36);(2,4)(5,8,12,22)(6,14,11,21)(7,16,17,30)(9,19,23,32)(10,20,25,34)(13,27,26,35)(15,28,29,36)(18,31,24,33);(5,9)(6,10)(7,17)(8,19)(11,25)(12,23)(14,20)(15,24)(16,30)(18,29)(21,34)(22,32)(28,33)(31,36)Polycyclic presentation:
2^8|b^2=h,c^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×#33834, #7623585, #7623607, … |
| maximal normal subgroup of | C2×#33834, #7623585, #7623607, … |
| maximal quotient of | #491733, #491734, #491745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26822);