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 | |
335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,8)(7,15,19,34)(9,22,26,33)(10,21,24,14)(12,29,16,18)(13,28,20,25)(17,35,27,31)(23,32,36,30);(5,7,11,19)(6,12,8,16)(9,23,26,36)(10,27,24,17)(13,31,20,35)(14,32,21,30)(15,33,34,22)(18,28,29,25);(2,4)(5,8,11,6)(7,16,19,12)(9,20,26,13)(10,21,24,14)(15,18,34,29)(17,32,27,30)(22,25,33,28)(23,35,36,31);(5,9)(6,13)(7,17)(8,20)(10,28)(11,26)(12,30)(14,33)(15,34)(16,32)(18,29)(19,27)(21,22)(24,25)Polycyclic presentation:
2^8|a^2=h,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×#33835, #7623585, #7623607, … |
| maximal normal subgroup of | C2×#33835, #7623585, #7623607, … |
| maximal quotient of | #491740, #491743, #491749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26823);