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 | #3285 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
332 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)(6,11)(7,15)(8,19)(9,22)(12,29)(13,33)(14,28)(16,32)(17,30)(18,25)(20,24)(23,26)(27,31)(34,35);(5,6,10,14)(7,16,18,29)(8,20,23,33)(9,24,22,13)(11,26,28,19)(12,30,32,17)(15,35,25,27)(21,34,36,31);(2,4);(5,7,10,18)(6,12,14,32)(8,21,23,36)(9,25,22,15)(11,27,28,35)(13,34,24,31)(16,33,29,20)(17,26,30,19)Polycyclic presentation:
2^8|b^2=h,d^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×#33836, #7623585, #7623608, … |
| maximal normal subgroup of | C2×#33836, #7623585, #7623608, … |
| maximal quotient of | #491731, #491738, #491754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26824);